# On the integral representation and the Raşa, Jensen and Hermite-Hadamard
inequalities for box-$(n_{1},\ldots,n_{d})$-convex functions

Andrzej Komisarski  Faculty of Mathematics and Computer Science, University of
Lodz, Banacha 22, 90–238 Łódź, Poland<EMAIL_ADDRESS>and  Teresa
Rajba  Department of Mathematics, University of Bielsko-Biala, ul. Willowa 2,
43-309 Bielsko-Biała, Poland<EMAIL_ADDRESS>
###### Abstract.

We give the integral representation of box-$(n_{1},\ldots,n_{d})$-convex
functions, which are a generalization of Popoviciu’s [14] $n$-convex and
box-$(m,n)$-convex functions. Based on this integral representation, we obtain
the Raşa, Jensen and Hermite-Hadamard inequalities for these functions. We
extend the results obtained in [4, 6, 7, 12, 14]

###### Key words and phrases:

divided differences and inequalities related to stochastic convex orderings
and box-convex functions and Hermite-Hadamard and Jensen inequalities and
functional inequalities related to convexity

###### 1991 Mathematics Subject Classification:

26D05 and 39B62 and 60E15

##  1\. Introduction

H. Hopf [8] and T. Popoviciu [14, 15] introduced the notion of higher order
convex functions based on the so called divided differences. Let $n\geq 1$.
Let $I$ be a subinterval of ${\mathbb{R}}$. Given a function $f\colon
I\to\mathbb{R}$ of one real variable and a system $x_{0},x_{1},\ldots,x_{n+1}$
of pairwise distinct points of $I$ the _divided differences_ of order
$0,1,\ldots,n+1$ are respectively defined by the formulas

$\displaystyle\left[x_{0};f\right]$ $\displaystyle=f(x_{0}),$
$\displaystyle\left[x_{0},x_{1};f\right]$
$\displaystyle=\frac{f(x_{1})-f(x_{0})}{x_{1}-x_{0}},\quad\ldots,$
$\displaystyle\left[x_{0},x_{1},\ldots,x_{n+1};f\right]$
$\displaystyle=\frac{\left[x_{1},x_{2},\ldots,x_{n+1};f\right]-\left[x_{0},x_{1},\ldots,x_{n};f\right]}{x_{n+1}-x_{0}}.$

The function $f(x)$ is called _$n$ -convex (resp. $n$-concave, $n$-affine)_,
if $[x_{0},x_{1},\ldots,$ $x_{n+1};f]\geq 0$ (resp. $\leq 0$, $=0$) for any
pairwise distinct points $x_{0},x_{1},\ldots,x_{n+1}$. The function $f$ is
convex when all divided differences of order two are nonnegative for all
systems of pairwise distinct points. If $f^{(n+1)}$ exists, then $f$ is _$n$
-convex ($n$-concave)_ if, and only if, $f^{(n+1)}\geq 0$ ($f^{(n+1)}\leq 0$).

######  Proposition 1 ([14]).

A function $f(x)$ is $n$-convex if and only if its derivative $f^{(n-1)}(x)$
exists and is convex (with the convention $f^{(0)}(x)=f(x)$).

If $f=f(x,y)$ is a function defined on a rectangle $I\times J$ and
$x_{0},x_{1},\ldots,x_{m}$ are pairwise distinct points of $I$ and
$y_{0},y_{1},\ldots,y_{n}$ are pairwise distinct points of $J$, one defines
the _divided double difference_ of order $(m,n)$ by the formula

(1) $\displaystyle\left[\LARGE{\begin{subarray}{c}{x_{0},\quad
x_{1},\quad\ldots\quad x_{m}}\\\ \\\ {y_{0},\quad y_{1},\quad\ldots\quad
y_{n}}\end{subarray}};f\right]$
$\displaystyle=\left[x_{0},x_{1},\ldots,x_{m};\left[y_{0},y_{1},\ldots,y_{n};f(x,\cdot)\right]\right]$
(2)
$\displaystyle=\left[y_{0},y_{1},\ldots,y_{n};\left[x_{0},x_{1},\ldots,x_{m};f(\cdot,y)\right]\right]$

Drawing a parallel to the one dimensional case, T. Popoviciu [14], p.78, has
called a function $f\colon I\times J\to\mathbb{R}$ _convex of order $(m,n)$_
(_box- $(m,n)$-convex_ in our terminology) if all divided differences

$\left[\LARGE{\begin{subarray}{c}{x_{0},\quad x_{1},\quad\ldots\quad x_{m}}\\\
\\\ {y_{0},\quad y_{1},\quad\ldots\quad y_{n}}\end{subarray}};f\right]$

are nonnegative for all pairwise distinct points $x_{0},x_{1},\ldots,x_{m}$
and $y_{0},y_{1},\ldots,y_{n}$. The related notions of _box- $(m,n)$-concave_
function and _box- $(m,n)$-affine_ function can be introduced in the standard
way.

In [12], we obtained the integral representation, the Raşa, Jensen and
Hermite-Hadamard inequalities for box-$(m,n)$-convex functions.

Let $\textbf{n}=(n_{1},\ldots,n_{d})$, $d\geq 1$,
$n_{1},\ldots,n_{d}\in\mathbb{N}$. In this paper, we define and study box-n-
convex functions of $d$ real variables, such that in the case $d=2$,
$n_{1}=m$, $n_{2}=n$, the class of box-n-convex functions coincides with the
class of box-$(m,n)$-convex functions. We give the integral representation and
several characterizations of box-n-convex functions. Using the integral
representation, we obtain the Raşa, Jensen and Hermite-Hadamard inequalities
for box-n-convex functions.

The results known in the literature regarding box-convex functions used
differential methods to study these functions. They required assumptions about
high regularity (the existence of derivatives of high orders). In this
article, we do not make any assumptions about regularity of the studied
functions (they may even be non-measurable). However, we still use
differential methods. This is possible thanks to the notion of
$\mathbf{n}$-regularity introduced in Section 5 and the regularization of
box-$\mathbf{n}$-convex functions, based on Proposition 6. We first used this
idea in [12] for functions of two variables. However, the general case (with
$d>2$) is much more complicated and introduces some phenomena that do not
occur for $d=2$.

In Section 2, we recall the selected properties of divided differences of
order $n$ given in [12].

In Section 3, we give the definitions and selected properties of multiple
divided differences of order $n$.

In Section 4, we prove that the set of box-n-affine functions coincides with
the set of pseudopolynomials of degree $(n_{1}-1,\ldots,n_{d}-1)$.

In Section 5, we introduce the notion of n-regularity of functions and we give
properties of n-regular functions.

In Section 6, we study properties of integration and differentation of
n-regular functions.

In Section 7, we obtain the integral representation of box-monotone function,
i.e. box-$(1,\ldots,1)$-convex functions.

In Section 8, we obtain the integral representation of box-n-convex function.

In Section 9, we give properties of box-n-convex orders.

In Section 10, we obtain the Hermite-Hadamard, Jensen and Raşa inequalities
for box-n-convex functions.

Throughout this article, we assume that $\mathbb{N}=\\{0,1,2,\dots\\}$ unless
otherwise stated.

##  2\. Selected properties of divided differences of order $n$

The following expanded form of divided difference is well known.

######  Proposition 2.

(3)
$\left[x_{0},x_{1},\ldots,x_{n};f\right]=\sum_{j=0}^{n}\frac{f(x_{j})}{\prod\limits_{\begin{subarray}{c}l=0\\\
l\neq j\end{subarray}}^{n}(x_{j}-x_{l})}.$

######  Lemma 1.

Let $n\geq 1$, let $I\subset\mathbb{R}$ be an interval, $\alpha\in I$ and let
$f\colon I\to\mathbb{R}$ be an integrable function. For pairwise distinct
points $x_{0},x_{1},\dots,x_{n}\in I$, we have

$\left[x_{0},x_{1},\dots,x_{n};\int_{\alpha}^{\cdot}f(t)dt\right]=\int_{0}^{1}t^{n-1}[x_{1,t},\dots,x_{n,t};f]dt,$

where $x_{i,t}=tx_{i}+(1-t)x_{0}$ for $t\in[0,1]$ and $i=1,2,\dots,n$.

###### Proof.

The proof of the lemma is by induction on $n$. For $n=1$ we have

$\left[x_{0},x_{1};\int_{\alpha}^{\cdot}f(t)dt\right]=\frac{\int_{x_{0}}^{x_{1}}f(t)dt}{x_{1}-x_{0}}=\int_{0}^{1}f(x_{1,t})dt=\int_{0}^{1}t^{1-1}[x_{1,t};f]dt.$

In the induction step, we use the definition of the divided difference and the
fact that it does not depend on the permutation of the points
$x_{0},\dots,x_{n+1}$.

$\left[x_{0},x_{1},\dots,x_{n+1};\int_{\alpha}^{\cdot}f(t)dt\right]=\left[x_{1},x_{0},x_{2},\dots,x_{n+1};\int_{\alpha}^{\cdot}f(t)dt\right]\\\
=\frac{\left[x_{0},x_{2},\dots,x_{n+1};\int_{\alpha}^{\cdot}f(t)dt\right]-\left[x_{0},x_{1},\dots,x_{n};\int_{\alpha}^{\cdot}f(t)dt\right]}{x_{n+1}-x_{1}}\\\
=\int_{0}^{1}t^{n-1}\frac{[x_{2,t},\dots,x_{n+1,t};f]-[x_{1,t},\dots,x_{n,t};f]}{x_{n+1}-x_{1}}dt\\\
=\int_{0}^{1}t^{n}\frac{[x_{2,t},\dots,x_{n+1,t};f]-[x_{1,t},\dots,x_{n,t};f]}{x_{n+1,t}-x_{1,t}}dt\\\
=\int_{0}^{1}t^{(n+1)-1}[x_{1,t},\dots,x_{n+1,t};f]dt.$

∎

######  Lemma 2.

Let $n\geq 2$, let $I\subset\mathbb{R}$ be an interval, and let $f\colon
I\to\mathbb{R}$ be a right-differentiable function. Let $f^{\prime}_{R}\colon
I\setminus\\{\sup I\\}\to\mathbb{R}$ be a right-derivative of $f$. For
pairwise distinct points $x_{1},\dots,x_{n}\in I\setminus\\{\sup I\\}$ and
$k=1,\dots,n$, the limit $\lim_{x_{0}\downarrow
x_{k}}[x_{0},x_{1},\dots,x_{n};f]$ exists and

(4) $\lim_{x_{0}\downarrow
x_{k}}[x_{0},x_{1},\dots,x_{n};f]=\frac{f^{\prime}_{R}(x_{k})}{\prod\limits_{\begin{subarray}{c}i=1\\\
i\neq k\end{subarray}}^{n}(x_{k}-x_{i})}+\sum_{\begin{subarray}{c}j=1\\\ j\neq
k\end{subarray}}^{n}\frac{1}{x_{j}-x_{k}}\left(\frac{f(x_{j})}{\prod\limits_{\begin{subarray}{c}i=1\\\
i\neq
j\end{subarray}}^{n}(x_{j}-x_{i})}+\frac{f(x_{k})}{\prod\limits_{\begin{subarray}{c}i=1\\\
i\neq k\end{subarray}}^{n}(x_{k}-x_{i})}\right).$

Moreover,

$[x_{1},\dots,x_{n};f^{\prime}_{R}]=\sum_{k=1}^{n}\lim_{x_{0}\downarrow
x_{k}}[x_{0},x_{1},\dots,x_{n};f].$

###### Proof.

Due to symmetry, it is enough to show that (4) holds for $k=1$. We skip an
easy proof by induction on $n$. Using (4), we obtain

$\sum_{k=1}^{n}\lim_{x_{0}\downarrow
x_{k}}[x_{0},x_{1},\dots,x_{n};f]=\sum_{k=1}^{n}\frac{f^{\prime}_{R}(x_{k})}{\prod\limits_{\begin{subarray}{c}i=1\\\
i\neq k\end{subarray}}^{n}(x_{k}-x_{i})}=[x_{1},\dots,x_{n};f^{\prime}_{R}].$

∎

##  3\. Divided differences of order $(n_{1},\ldots,n_{d})$ and
box-$(n_{1},\ldots,n_{d})$-convex functions

Let $d\in\mathbb{N}$. For every $A\subset\\{1,2,\dots,d\\}$ let
$A^{\prime}=\\{1,2,\dots,d\\}\setminus A$ be the complement of the set $A$.

For $\mathbf{x}\in\mathbb{R}^{d}$, $\mathbf{n}\in\mathbb{N}^{d}$ and
$A\subset\\{1,2,\dots,d\\}$ let
$\mathbf{x}_{A}=(x_{a_{1}},x_{a_{2}},\dots,x_{a_{|A|}})$, and
$\mathbf{n}_{A}=(n_{a_{1}},n_{a_{2}},\dots,n_{a_{|A|}})$, where
$(a_{1},a_{2},\dots,a_{|A|})$ is the ordered sequence of the elements of $A$
(i.e., $A=\\{a_{1},a_{2},\dots,a_{|A|}\\}$ and $a_{1}<a_{2}<\dots<a_{|A|}$).
In particular $\mathbf{x}_{\\{i\\}}=(x_{i})$ and
$\mathbf{x}_{\\{i\\}^{\prime}}=(x_{1},\dots,x_{i-1},$ $x_{i+1},\dots,x_{d})$.

Let $I_{1},I_{2},\dots,I_{d}\subset\mathbb{R}$ be intervals (bounded or
unbounded). We denote $\mathbf{I}=\prod_{i=1}^{d}I_{i}\subset\mathbb{R}^{d}$.
If $A\subset\\{1,2,\dots,d\\}$, then we put $\mathbf{I}_{A}=\prod_{i\in
A}I_{i}\subset\mathbb{R}^{|A|}$. Clearly, if $\mathbf{x}\in\mathbf{I}$, then
$\mathbf{x}_{A}\in\mathbf{I}_{A}$.

For $f\colon\mathbf{I}\to\mathbb{R}$, $A\subset\\{1,2,\dots,d\\}$ and
$\mathbf{z}\in\mathbf{I}_{A^{\prime}}$ we denote by
$f_{A}^{\mathbf{z}}=f_{A}\colon\mathbf{I}_{A}\to\mathbb{R}$ the function given
by

$f_{A}^{\mathbf{z}}(\mathbf{y})=f(\mathbf{x}),$

where

$x_{i}=\begin{cases}y_{j}&\text{if $i\in A$ and }a_{j}=i,\\\ z_{j}&\text{if
$i\in A^{\prime}$ and }a^{\prime}_{j}=i,\end{cases}$

$(a_{1},a_{2},\dots,a_{|A|})$ is the ordered sequence of the elements of $A$,
and $(a^{\prime}_{1},a^{\prime}_{2},$ $\dots,a^{\prime}_{|A^{\prime}|})$ is
the ordered sequence of the elements of $A^{\prime}$ (i.e., $\mathbf{x}$ is
the unique element of $\mathbf{I}$ satisfying $\mathbf{x}_{A}=\mathbf{y}$ and
$\mathbf{x}_{A^{\prime}}=\mathbf{z}$). In particular,
$f_{A}^{\mathbf{x}_{A^{\prime}}}(\mathbf{x}_{A})=f(\mathbf{x})$. The idea
behind the notation $f_{A}^{\cdot}(\cdot)$ is to reduce the number of
variables of $f$ by treating some of them as parameters (those that are
standing at the positions described by the set $A^{\prime}$).

It is convenient to consider the special case $d=0$. Then
$f\colon\mathbf{I}\to\mathbb{R}$ is a constant (a function of $0$ variables).

######  Definition 1.

Let $f\colon\mathbf{I}\to\mathbb{R}$ and
$\mathbf{n}=(n_{1},n_{2},\dots,n_{d})\in\mathbb{N}^{d}$. For $i=1,2,\dots,d$
let $\mathbf{x}_{i}=(x_{i0},x_{i1},\dots,x_{in_{i}})\in I_{i}^{n_{i}+1}$ be a
vector with pairwise distinct coordinates. We inductively define the multiple
divided difference of order $\mathbf{n}$ as follows.

If $d=0$ (i.e. $f$ is a constant), then $[;f]=f$.

If $d=1$, then $[\mathbf{x}_{1};f]=[x_{10},x_{11},\dots,x_{1n_{1}};f]$ is the
divided difference defined in the Introduction (Section 1).

If $d>1$, then

$\left[\ \LARGE{\begin{subarray}{c}\mathbf{x}_{1}\\\ \\\ \mathbf{x}_{2}\\\ \\\
\dots\\\ \\\
\mathbf{x}_{d}\end{subarray}};f\right]=\left[\LARGE{\begin{subarray}{c}x_{10},\
x_{11},\ldots,\ x_{1n_{1}}\\\ \\\ x_{20},\ x_{21},\ldots,\ x_{2n_{2}}\\\ \\\
\dots\\\ \\\ x_{d0},\ x_{d1},\ldots,\
x_{dn_{d}}\end{subarray}};f\right]=[x_{d0},x_{d1},\ldots,\ x_{dn_{d}};g],$

where $g\colon I_{d}\to\mathbb{R}$ is given by

$g(x)=\left[\LARGE{\begin{subarray}{c}x_{10},\ x_{11},\ldots,\ x_{1n_{1}}\\\
\\\ x_{20},\ x_{21},\ldots,\ x_{2n_{2}}\\\ \\\ \dots\\\ \\\ x_{d-1,0},\
x_{d-1,1},\ldots,\
x_{d-1,n_{d-1}}\end{subarray}};f_{\\{1,2,\dots,d-1\\}}^{x}\right].$

In other words, we apply the divided differences to successive arguments of
$f$.

In the following proposition, we give the expanded form of divided difference
of order n, which is a generalization of (3).

######  Proposition 3.

If $d>0$, then

(5) $\left[\LARGE{\begin{subarray}{c}x_{10},\ x_{11},\ldots,\ x_{1n_{1}}\\\
\\\ x_{20},\ x_{21},\ldots,\ x_{2n_{2}}\\\ \\\ \dots\\\ \\\ x_{d0},\
x_{d1},\ldots,\
x_{dn_{d}}\end{subarray}};f\right]=\sum_{j_{1}=0}^{n_{1}}\sum_{j_{2}=0}^{n_{2}}\dots\sum_{j_{d}=0}^{n_{d}}\frac{f(x_{1j_{1}},x_{2j_{2}},\dots,x_{dj_{d}})}{\prod\limits_{i=1}^{d}\prod\limits_{\begin{subarray}{c}l_{i}=0\\\
l_{i}\neq j_{i}\end{subarray}}^{n_{i}}(x_{ij_{i}}-x_{il_{i}})}.$

###### Sketch of the proof.

By induction on $m=1,2,\dots,d$ and taking into account Proposition 2, we
obtain

$\left[\LARGE{\begin{subarray}{c}x_{10},\ x_{11},\ldots,\ x_{1n_{1}}\\\ \\\
x_{20},\ x_{21},\ldots,\ x_{2n_{2}}\\\ \\\ \dots\\\ \\\ x_{m0},\
x_{m1},\ldots,\
x_{mn_{m}}\end{subarray}};g\right]=\sum_{j_{1}=0}^{n_{1}}\sum_{j_{2}=0}^{n_{2}}\dots\sum_{j_{m}=0}^{n_{m}}\frac{g(x_{1j_{1}},x_{2j_{2}},\dots,x_{mj_{m}})}{\prod\limits_{i=1}^{m}\prod\limits_{\begin{subarray}{c}l_{i}=0\\\
l_{i}\neq j_{i}\end{subarray}}^{n_{i}}(x_{ij_{i}}-x_{il_{i}})},$

for every function $g\colon\mathbf{I}_{\\{1,2,\dots,m\\}}\to\mathbb{R}$ such
that $g=f_{\\{1,2,\dots,m\\}}^{\mathbf{y}}$ for some $\mathbf{y}\in
I_{\\{m+1,\dots,d\\}}$. For $m=d$ we obtain (5). ∎

By symmetry of formula (5), we see that the order in which the divided
differences are applied in Definition 1 is not important (similarly, as in the
two-dimensional case, cf. [4, 12, 14]). In particular, the following
observation holds.

######  Remark 3.

If $A\subset\\{1,2,\dots,d\\}$, $A=\\{a_{1},a_{2},\dots,a_{|A|}\\}$,
$a_{1}<a_{2}<\dots<a_{|A|}$, and
$A^{\prime}=\\{a^{\prime}_{1},a^{\prime}_{2},\dots,a^{\prime}_{|A^{\prime}|}\\}$,
$a^{\prime}_{1}<a^{\prime}_{2}<\dots<a^{\prime}_{|A^{\prime}|}$, then

$\left[\ \LARGE{\begin{subarray}{c}\mathbf{x}_{1}\\\ \\\ \mathbf{x}_{2}\\\ \\\
\dots\\\ \\\ \mathbf{x}_{d}\end{subarray}};f\right]=\left[\
\LARGE{\begin{subarray}{c}\mathbf{x}_{a^{\prime}_{1}}\\\ \\\
\mathbf{x}_{a^{\prime}_{2}}\\\ \\\ \dots\\\ \\\
\mathbf{x}_{a^{\prime}_{|A^{\prime}|}}\end{subarray}};g\right],$

where $g\colon\mathbf{I}_{A^{\prime}}\to\mathbb{R}$ is given by

$g(\mathbf{x})=\left[\LARGE{\begin{subarray}{c}\mathbf{x}_{a_{1}}\\\ \\\
\mathbf{x}_{a_{2}}\\\ \\\ \dots\\\ \\\
\mathbf{x}_{a_{|A|}}\end{subarray}};f_{A}^{\mathbf{x}}\right].$

An immediate consequence of Remark 3 is the following observation.

######  Remark 4.

Let $A\subset\\{1,2,\dots,d\\}$, $A=\\{a_{1},a_{2},\dots,a_{|A|}\\}$,
$a_{1}<a_{2}<\dots<a_{|A|}$, and
$A^{\prime}=\\{a^{\prime}_{1},a^{\prime}_{2},\dots,a^{\prime}_{|A^{\prime}|}\\}$,
$a^{\prime}_{1}<a^{\prime}_{2}<\dots<a^{\prime}_{|A^{\prime}|}$. Let
$f\colon\mathbf{I}\to\mathbb{R}$ be the function given by the formula
$f(\mathbf{x})=g(\mathbf{x}_{A})\cdot h(\mathbf{x}_{A^{\prime}})$, where
$g\colon\mathbf{I}_{A}\to\mathbb{R}$ and
$h\colon\mathbf{I}_{A^{\prime}}\to\mathbb{R}$. Then

$\left[\ \LARGE{\begin{subarray}{c}\mathbf{x}_{1}\\\ \\\ \mathbf{x}_{2}\\\ \\\
\dots\\\ \\\ \mathbf{x}_{d}\end{subarray}};f\right]=\left[\
\LARGE{\begin{subarray}{c}\mathbf{x}_{a_{1}}\\\ \\\ \mathbf{x}_{a_{2}}\\\ \\\
\dots\\\ \\\ \mathbf{x}_{a_{|A|}}\end{subarray}};g\right]\cdot\left[\
\LARGE{\begin{subarray}{c}\mathbf{x}_{a^{\prime}_{1}}\\\ \\\
\mathbf{x}_{a^{\prime}_{2}}\\\ \\\ \dots\\\ \\\
\mathbf{x}_{a^{\prime}_{|A^{\prime}|}}\end{subarray}};h\right].$

######  Definition 2.

Let $\mathbf{n}\in\mathbb{N}^{d}$. We say that the function
$f\colon\mathbf{I}\to\mathbb{R}$ is _box- $\mathbf{n}$-convex_ if

(6) $\left[\ \LARGE{\begin{subarray}{c}\mathbf{x}_{1}\\\ \\\ \mathbf{x}_{2}\\\
\\\ \dots\\\ \\\ \mathbf{x}_{d}\end{subarray}};f\right]\geq 0$

for every $\mathbf{x}_{1},\mathbf{x}_{2},\dots,\mathbf{x}_{d}$, where
$\mathbf{x}_{i}=(x_{i0},x_{i1},\dots,x_{in_{i}})\in I_{i}^{n_{i}+1}$ is a
vector with pairwise distinct coordinates ($i=1,2,\dots,d$).

We say that $f$ is _box- $\mathbf{n}$-concave_ if it satisfies (6) with $\geq$
replaced by $\leq$.

We say that $f$ is _box- $\mathbf{n}$-affine_ if (6) becomes the equality.

######  Remark 5.

Observe that for every $\mathbf{n}\in\mathbb{N}^{d}$ the sets of all
box-$\mathbf{n}$-convex functions and all box-$\mathbf{n}$-concave functions
are convex cones in the space of all real functions on $\mathbf{I}$. The set
of all box-$\mathbf{n}$-affine functions is a linear subspace of that space.

######  Remark 6.

The case $d=2$ was investigated in [12].

If $f\colon I\to\mathbb{R}$ is a function of one variable ($d=1$), then $f$ is
box-$(n)$-convex if and only if it is $(n-1)$-convex in the classical sense.
In particular, box-$(0)$-convexity of a function $f$ means that $f$ is non-
negative. Box-$(1)$-convexity of $f$ means that $f$ is non-decreasing.
Box-$(2)$-convexity of a function is equivalent to its convexity. If $f\colon
I\to\mathbb{R}$ is box-$(2)$-convex and $I\subset\mathbb{R}$ is open, then $f$
is continuous and right-differentiable. For $n>2$ and open $I$ every
box-$(n)$-convex function has a derivative of order $n-2$, which is a convex
function.

By Remark 4 we obtain the following remark.

######  Remark 7.

Let $\mathbf{n}\in\mathbb{N}$ and $A\subset\\{1,2,\dots,d\\}$. Let
$f\colon\mathbf{I}\to\mathbb{R}$ be the function given by the formula
$f(\mathbf{x})=g(\mathbf{x}_{A})\cdot h(\mathbf{x}_{A^{\prime}})$, where
$g\colon\mathbf{I}_{A}\to\mathbb{R}$ and
$h\colon\mathbf{I}_{A^{\prime}}\to\mathbb{R}$. Then

  * (a)

if $g$ is box-$\mathbf{n}_{A}$-convex and $h$ is
box-$\mathbf{n}_{A^{\prime}}$-convex, then $f$ is box-$\mathbf{n}$-convex.

  * (b)

if $g$ is a linear combination of box-$\mathbf{n}_{A}$-convex functions and
$h$ is a linear combination of box-$\mathbf{n}_{A^{\prime}}$-convex functions,
then $f$ is a linear combination of box-$\mathbf{n}$-convex functions.

  * (c)

if $g$ is box-$\mathbf{n}_{A}$-affine or $h$ is
box-$\mathbf{n}_{A^{\prime}}$-affine, then $f$ is box-$\mathbf{n}$-affine.

##  4\. Pseudo-polynomials and box-$\mathbf{n}$-affine functions

######  Definition 3.

Let $\mathbf{n}=(n_{1},n_{2},\dots,n_{d})\in\\{-1,0,\dots\\}^{d}$. We say that
the function $W\colon\mathbf{I}\to\mathbb{R}$ is a _pseudo-polynomial of
degree_ $\mathbf{n}$ if it is of the form

$W(x_{1},x_{2},\dots,x_{d})=\sum_{i=1}^{d}\sum_{k=0}^{n_{i}}A_{ik}(\mathbf{x}_{\\{i\\}^{\prime}})x_{i}^{k},$

where $\mathbf{x}=(x_{1},x_{2},\dots,x_{d})$, and
$A_{ik}\colon\mathbf{I}_{\\{i\\}^{\prime}}\to\mathbb{R}$ ($i=1,2,\dots,d$ and
$k=0,1,\dots,n_{i}$) are arbitrary functions (as usually,
$\sum_{k=0}^{-1}A_{ik}(\mathbf{x}_{\\{i\\}^{\prime}})x_{i}^{k}=0$ if
$n_{i}=-1$).

######  Example 1.

Let $d=3$ and $\mathbf{n}=(n_{1},n_{2},n_{3})\in\mathbb{N}^{3}$. Then, the
function $W\colon I_{1}\times I_{2}\times I_{3}\to\mathbb{R}$ is a _pseudo-
polynomial_ of degree $\mathbf{n}$ if it is of the form

$W(x_{1},x_{2},x_{3})=\sum_{k=0}^{n_{1}}A_{1k}(x_{2},x_{3})x_{1}^{k}+\sum_{k=0}^{n_{2}}A_{2k}(x_{1},x_{3})x_{2}^{k}+\sum_{k=0}^{n_{3}}A_{3k}(x_{1},x_{2})x_{3}^{k},$

where $A_{1k}\colon I_{2}\times I_{3}\to\mathbb{R}$ $(k=0,1,\dots,n_{1})$,
$A_{2k}\colon I_{1}\times I_{3}\to\mathbb{R}$ $(k=0,1,\dots,n_{2})$ and
$A_{3k}\colon I_{1}\times I_{2}\to\mathbb{R}$ $(k=0,1,\dots,n_{3})$ are
arbitrary functions.

######  Remark 8.

Note that, in general, the degree of a pseudo-polynomial is not uniquely
determined. For example $W(x,y)=e^{x}y+e^{y}x+x^{2}y^{2}$ is a pseudo-
polynomial of both degrees $(1,2)$ and $(2,1)$, but it is not a pseudo-
polynomial of degree $(1,1)$.

######  Lemma 9.

Let $f\colon\mathbf{I}\to\mathbb{R}$. We fix $i\in\\{1,2,\dots,d\\}$. Let
$n_{i}\in\mathbb{N}$, and let $u_{i1},u_{i2},\dots,u_{in_{i}}$ be pairwise
distinct elements of $I_{i}$. There exists a pseudo-polynomial
$W(x_{1},x_{2},\dots,x_{d})=\sum_{k=0}^{n_{i}-1}A_{ik}(\mathbf{x}_{\\{i\\}^{\prime}})x_{i}^{k}$
such that

$f(x_{1},x_{2},\dots,x_{d})=W(x_{1},x_{2},\dots,x_{d}),$

whenever $x_{i}\in\\{u_{i1},u_{i2},\dots,u_{in_{i}}\\}$.

Moreover, for each $k=0,1,\dots,n_{i}-1$, the function $A_{ik}$ is a linear
combination of the functions $f_{\\{i\\}^{\prime}}^{u_{i1}}$,
$f_{\\{i\\}^{\prime}}^{u_{i2}}$, …, $f_{\\{i\\}^{\prime}}^{u_{in_{i}}}$, where

$f_{\\{i\\}^{\prime}}^{u_{ij}}(x_{1},\dots,x_{i-1},x_{i+1},\dots,x_{d})=f(x_{1},\dots,x_{i-1},u_{ij},x_{i+1},\dots,x_{d})$

for $j=1,2,\dots,n_{i}$.

###### Proof.

We fix $i\in\\{1,2,\dots,d\\}$. For every
$\mathbf{y}\in\mathbf{I}_{\\{i\\}^{\prime}}$, let
$A_{ik}(\mathbf{y})\in\mathbb{R}$ $(k=0,1,\ldots,n_{i}-1)$ be such that
$L(x)=L_{\mathbf{y}}(x)=\sum_{k=0}^{n_{i}-1}A_{ik}(\mathbf{y})x^{k}$ is the
Lagrange interpolation polynomial of the function $f_{\\{i\\}}^{\mathbf{y}}$
for $n_{i}$ nodes $u_{i1},u_{i2},\dots,u_{in_{i}}$. Then
$L_{\mathbf{y}}(x)=f(y_{1},\dots,y_{i-1}$, $x$, $y_{i+1},\dots,y_{d})$ for
$x\in\\{u_{i1},u_{i2},\dots,u_{in_{i}}\\}$, and the coefficients
$A_{ik}(\mathbf{y})$ are linear combinations of the values
$f_{\\{i\\}}^{\mathbf{y}}(u_{i1})=f_{\\{i\\}^{\prime}}^{u_{i1}}(\mathbf{y}),\dots,f_{\\{i\\}}^{\mathbf{y}}(u_{in_{i}})=f_{\\{i\\}^{\prime}}^{u_{in_{i}}}(\mathbf{y})$.
We put

$W(x_{1},x_{2},\dots,x_{d})=L_{\mathbf{x}_{\\{i\\}^{\prime}}}(x_{i})=\sum_{k=0}^{n_{i}-1}A_{ik}(\mathbf{x}_{\\{i\\}^{\prime}})x_{i}^{k}.$

The lemma is proved. ∎

######  Lemma 10.

Let $f\colon\mathbf{I}\to\mathbb{R}$,
$\mathbf{n}=(n_{1},n_{2},\dots,n_{d})\in\mathbb{N}^{d}$. For each
$i=1,2,\dots,d$, let $u_{i1},u_{i2},\dots,u_{in_{i}}$ be pairwise distinct
elements of $I_{i}$. There exists a pseudo-polynomial $W$ of degree
$(n_{1}-1,n_{2}-1,\dots,n_{d}-1)$, such that

(7)
$f(x_{1},\dots,x_{i-1},u_{ij},x_{i+1},\dots,x_{d})=W(x_{1},\dots,x_{i-1},u_{ij},x_{i+1},\dots,x_{d})$

for each $i=1,2,\dots,d$ and $j=1,2,\dots,n_{i}$.

###### Proof.

We will define functions $g_{i}\colon\mathbf{I}\to\mathbb{R}$ and pseudo-
polynomials $W_{i}\colon\mathbf{I}\to\mathbb{R}$ (for $i=1,2,\dots,d$) by
induction. First we put $g_{i}=f-\sum_{j=1}^{i-1}W_{j}$. Next, the pseudo-
polynomial

$W_{i}(x_{1},x_{2},\dots,x_{d})=\sum_{k=0}^{n_{i}-1}A_{ik}(\mathbf{x}_{\\{i\\}^{\prime}})x_{i}^{k}$

such that $g_{i}(x_{1},x_{2},\dots,x_{d})=W_{i}(x_{1},x_{2},\dots,x_{d})$,
whenever $x_{i}\in\\{u_{i1},u_{i2},\dots$, $u_{in_{i}}\\}$, is obtained using
Lemma 9.

Taking the pseudo-polynomial $W=\sum_{i=1}^{d}W_{i}$, we obtain that equation
(7) is satisfied. The lemma is proved. ∎

######  Lemma 11.

Let $\mathbf{n}=(n_{1},n_{2},\dots,n_{d})\in\mathbb{N}^{d}$. If
$W\colon\mathbf{I}\to\mathbb{R}$ is a pseudo-polynomial of degree
$(n_{1}-1,n_{2}-1,\dots,n_{d}-1)$, then $W$ is box-$\mathbf{n}$-affine.

###### Proof.

Let $W\colon\mathbf{I}\to\mathbb{R}$ be a pseudo-polynomial of degree
$(n_{1}-1,n_{2}-1,\dots,n_{d}-1)$ of the form

$W(\mathbf{x})=\sum_{i=1}^{d}\sum_{k=0}^{n_{i}-1}A_{ik}(\mathbf{x}_{\\{i\\}^{\prime}})x_{i}^{k},$

where $A_{ik}\colon\mathbf{I}_{\\{i\\}^{\prime}}\to\mathbb{R}$
($i=1,2,\dots,d$ and $k=0,1,\dots,n_{i}-1$) are arbitrary functions.

For a fixed $i=1,2,\dots,d$ and $k=0,1,\dots,n_{i}-1$ we have that $I_{i}\ni
x_{i}\mapsto x_{i}^{k}$ is a box-$(n_{i})$-affine function and
$A_{ik}\colon\mathbf{I}_{\\{i\\}^{\prime}}\to\mathbb{R}$. By Remark 7(c) we
obtain that $\mathbf{I}\ni\mathbf{x}\mapsto
A_{ik}(\mathbf{x}_{\\{i\\}^{\prime}})x_{i}^{k}$ is box-$\mathbf{n}$-affine.
Consequently,
$W(\mathbf{x})=\sum_{i=1}^{d}\sum_{k=0}^{n_{i}-1}A_{ik}(\mathbf{x}_{\\{i\\}^{\prime}})x_{i}^{k}$
is box-$\mathbf{n}$-affine. ∎

######  Proposition 4.

Let $\mathbf{n}\in\mathbb{N}^{d}$ and $f,g\colon\mathbf{I}\to\mathbb{R}$ be
two box-$\mathbf{n}$-affine functions. Let $u_{i1},u_{i2},\dots,u_{in_{i}}$ be
pairwise distinct elements of $I_{i}$ ($i=1,2,\dots,d$). If

(8)
$f(x_{1},\dots,x_{i-1},u_{ij},x_{i+1},\dots,x_{d})=g(x_{1},\dots,x_{i-1},u_{ij},x_{i+1},\dots,x_{d})$

for each $i=1,2,\dots,d$ and $j=1,2,\dots,n_{i}$, then $f=g$.

###### Proof.

Let $h=f-g$. For $i=1,2,\dots,d$ let $x_{i0}=x_{i}$ and $x_{ij}=u_{ij}$
($j=1,2,\dots,n_{i}$). Since $h$ is box-$\mathbf{n}$-affine, by Proposition 3
we obtain

$\sum_{j_{1}=0}^{n_{1}}\sum_{j_{2}=0}^{n_{2}}\dots\sum_{j_{d}=0}^{n_{d}}\frac{h(x_{1j_{1}},x_{2j_{2}},\dots,x_{dj_{d}})}{\prod\limits_{i=1}^{d}\prod\limits_{\begin{subarray}{c}l_{i}=0\\\
l_{i}\neq j_{i}\end{subarray}}^{n_{i}}(x_{ij_{i}}-x_{il_{i}})}=0.$

Equality (8) implies that $h(x_{1j_{1}},x_{2j_{2}},\dots,x_{dj_{d}})=0$,
whenever at least one of $j_{i}$’s is non-zero. Consequently,
$h(x_{1},x_{2},\dots,x_{d})=h(x_{10},x_{20},\dots,x_{d0})=0$. ∎

######  Theorem 12.

Let $\mathbf{n}=(n_{1},n_{2},\dots,n_{d})\in\mathbb{N}^{d}$. Then a function
$f\colon\mathbf{I}\to\mathbb{R}$ is box-$\mathbf{n}$-affine, if and only if it
is a pseudo-polynomial of degree $(n_{1}-1,n_{2}-1,\dots,n_{d}-1)$.

###### Proof.

For the implication ($\Leftarrow$) see Lemma 11. We need to show the
implication ($\Rightarrow$).

For each $i=1,2,\dots,d$, we fix arbitrary $n_{i}$ pairwise distinct points
$u_{i1}$, $u_{i2}$, …, $u_{in_{i}}\in I_{i}$. Let $W$ be the pseudo-polynomial
of degree $(n_{1}-1,n_{2}-1,\dots,n_{d}-1)$ obtained by Lemma 10, for which
(7) is satisfied. By Lemma 11, we have that $W$ is box-$\mathbf{n}$-affine.
Then, by Proposition 4 it follows that $f=W$. The theorem is proved. ∎

##  5\. $\mathbf{n}$-regular functions

The box-$\mathbf{n}$-convex functions may be very irregular (they may even be
non-measurable, e.g. $f(x_{1},\dots,x_{d})=h(x_{1})$ is
box-$\mathbf{n}$-convex for each $d\geq 2$,
$\mathbf{n}\in\mathbb{\\{}1,2,\dots\\}^{d}$ and any function $h$ of one
variable). However, we will use differential methods to study them. This is
possible thanks to the notion of $\mathbf{n}$-regularity introduced in this
section.

######  Definition 4.

Let $f\colon\mathbf{I}\to\mathbb{R}$ and $\mathbf{n}\in\mathbb{N}^{d}$. We say
that the function $f$ is $\mathbf{n}$-regular if the functions
$f_{A}^{\mathbf{z}}\colon\mathbf{I}_{A}\to\mathbb{R}$ are linear combinations
of box-$\mathbf{n}_{A}$-convex functions for every $A\subset\\{1,2,\dots,d\\}$
and $\mathbf{z}\in\mathbf{I}_{A^{\prime}}$.

######  Remark 13.

Let $\mathbf{n}\in\mathbb{N}^{d}$.

  * (a)

The set of all $\mathbf{n}$-regular functions is a linear subspace of the
space of all real functions on $\mathbf{I}$.

  * (b)

If $f\colon\mathbf{I}\to\mathbb{R}$ is $\mathbf{n}$-regular,
$A\subset\\{1,2,\dots,d\\}$ and $\mathbf{z}\in\mathbf{I}_{A^{\prime}}$, then
$f_{A}^{\mathbf{z}}$ is $\mathbf{n}_{A}$-regular.

  * (c)

Let $A\subset\\{1,2,\dots,d\\}$ and let $f\colon\mathbf{I}\to\mathbb{R}$ be
the function given by the formula $f(\mathbf{x})=g(\mathbf{x}_{A})\cdot
h(\mathbf{x}_{A^{\prime}})$, where $g\colon\mathbf{I}_{A}\to\mathbb{R}$ and
$h\colon\mathbf{I}_{A^{\prime}}\to\mathbb{R}$. If $g$ is
$\mathbf{n}_{A}$-regular and $h$ is $\mathbf{n}_{A^{\prime}}$-regular, then
$f$ is $\mathbf{n}$-regular (by Remark 7(b)).

######  Lemma 14.

Let $\mathbf{n}\in\mathbb{N}^{d}$ and let a pseudo-polynomial
$W\colon\mathbf{I}\to\mathbb{R}$ of degree $\mathbf{n}$ be given by

$W(x_{1},x_{2},\dots,x_{d})=\sum_{i=1}^{d}\sum_{k=0}^{n_{i}}A_{ik}(\mathbf{x}_{\\{i\\}^{\prime}})x_{i}^{k}.$

If the functions $A_{ik}$ are $\mathbf{n}_{\\{i\\}^{\prime}}$-regular for each
$i=1,2,\dots,d$ and $k=0,1,\dots,n_{i}$, then the pseudo-polynomial $W$ is
$\mathbf{n}$-regular.

If $d>2$, then the converse theorem does not hold. See Remark 16.

###### Proof.

Let $i=1,2,\dots,d$ and $k=0,1,\dots,n_{i}$ be fixed. Since $I_{i}\ni
x_{i}\mapsto x_{i}^{k}$ is box-$(n_{i})$-convex and
$A_{ik}\colon\mathbf{I}_{\\{i\\}^{\prime}}\to\mathbb{R}$ is
$\mathbf{n}_{\\{i\\}^{\prime}}$-regular, we obtain (by Remark 13(c)) that the
function $\mathbf{I}\ni\mathbf{x}\mapsto
A_{ik}(\mathbf{x}_{\\{i\\}^{\prime}})x_{i}^{k}$ is $\mathbf{n}$-regular. This
implies that
$W(\mathbf{x})=\sum_{i=1}^{d}\sum_{k=0}^{n_{i}}A_{ik}(\mathbf{x}_{\\{i\\}^{\prime}})x_{i}^{k}$
is $\mathbf{n}$-regular. ∎

The following lemma is a straightening of Lemma 10 for $\mathbf{n}$-regular
functions.

######  Lemma 15.

Let $\mathbf{n}\in\mathbb{N}^{d}$ and let $f\colon\mathbf{I}\to\mathbb{R}$ be
an $\mathbf{n}$-regular function. For each $i=1,2,\dots,d$, let
$u_{i1},u_{i2},\dots,u_{i,n_{i}+1}$ be pairwise distinct elements of $I_{i}$.
Then for each $i=1,2,\dots,d$ there exist
$\mathbf{n}_{\\{i\\}^{\prime}}$-regular functions $A_{i0},A_{i1},\dots$,
$A_{in_{i}}\colon\mathbf{I}_{\\{i\\}^{\prime}}\to\mathbb{R}$ such that the
pseudo-polynomial

$W(x_{1},x_{2},\dots,x_{d})=\sum_{i=1}^{d}\sum_{k=0}^{n_{i}}A_{ik}(\mathbf{x}_{\\{i\\}^{\prime}})x_{i}^{k}$

is $\mathbf{n}$-regular, and

(9)
$f(x_{1},\dots,x_{i-1},u_{ij},x_{i+1},\dots,x_{d})=W(x_{1},\dots,x_{i-1},u_{ij},x_{i+1},\dots,x_{d})$

for each $i=1,2,\dots,d$ and $j=1,2,\dots,n_{i}+1$.

###### Proof.

Similarly to the proof of Lemma 10, we will define by induction functions
$g_{i}\colon\mathbf{I}\to\mathbb{R}$ and pseudo-polynomials
$W_{i}\colon\mathbf{I}\to\mathbb{R}$ for $i=1,2,\dots,d$. First we put
$g_{i}=f-\sum_{j=1}^{i-1}W_{j}$. Next, the pseudo-polynomial

$W_{i}(x_{1},x_{2},\dots,x_{d})=\sum_{k=0}^{n_{i}}A_{ik}(\mathbf{x}_{\\{i\\}^{\prime}})x_{i}^{k}$

such that $g_{i}(x_{1},x_{2},\dots,x_{d})=W_{i}(x_{1},x_{2},\dots,x_{d})$,
whenever $x_{i}\in\\{u_{i1},u_{i2},\dots$, $u_{i,n_{i}+1}\\}$ is obtained
using Lemma 9. (Note, that we use Lemma 9 with $n_{i}+1$ in place of $n_{i}$.)

By Lemma 9, Lemma 14 and Remark 13(b), we have that for each $i=1,2,\dots,d$
and $k=0,1,\dots,n_{i}$ the function $A_{ik}$ is
$\mathbf{n}_{\\{i\\}^{\prime}}$-regular as a linear combination of
$\mathbf{n}_{\\{i\\}^{\prime}}$-regular functions
$(g_{i})_{\\{i\\}^{\prime}}^{u_{i1}}$, $(g_{i})_{\\{i\\}^{\prime}}^{u_{i2}}$,
…, $(g_{i})_{\\{i\\}^{\prime}}^{u_{i,n_{i}+1}}$.

Taking the pseudo-polynomial $W=\sum_{i=1}^{d}W_{i}$, we obtain that equation
(9) is satisfied. By Lemma 14, $W$ is $\mathbf{n}$-regular. The proof is
completed. ∎

######  Proposition 5.

Let $\mathbf{n}\in\mathbb{N}^{d}$. A pseudo-polynomial
$W\colon\mathbf{I}\to\mathbb{R}$ of degree $\mathbf{n}$ is
$\mathbf{n}$-regular if and only if for each $i=1,2,\dots,d$ there exist
$\mathbf{n}_{\\{i\\}^{\prime}}$-regular functions
$A_{ik}\colon\mathbf{I}\to\mathbb{R}$ $(k=0,1,\dots,n_{i})$ such that

$W(x_{1},x_{2},\dots,x_{d})=\sum_{i=1}^{d}\sum_{k=0}^{n_{i}}A_{ik}(\mathbf{x}_{\\{i\\}^{\prime}})x_{i}^{k}.$

###### Proof.

For the implication ($\Leftarrow$) see Lemma 14. We need to show the
implication ($\Rightarrow$). For each $i=1,2,\dots,d$, we choose arbitrary
pairwise distinct $u_{i1},u_{i2},\dots$, $u_{i,n_{i}+1}\in I_{i}$. We apply
Lemma 15 for the function $W$ and we obtain the pseudo-polynomial
$W^{\prime}(x_{1},x_{2},\dots,x_{d})=\sum_{i=1}^{d}\sum_{k=0}^{n_{i}}A_{ik}(\mathbf{x}_{\\{i\\}^{\prime}})x_{i}^{k}$
with $\mathbf{n}_{\\{i\\}^{\prime}}$-regular coefficients $A_{ik}$. Since both
$W$ and $W^{\prime}$ are $(n_{1}+1,n_{2}+1,\dots,n_{d}+1)$-affine functions
(by Theorem 12) and they satisfy (9), we obtain (by Proposition 4) that
$W=W^{\prime}$. ∎

######  Remark 16.

According to Proposition 5, a pseudo-polynomial is regular if it has a
representation with regular coefficients. However, for $d\geq 3$, every
regular pseudo-polynomial has also representations with non-regular
coefficients. For example the $(1,1,1)$-regular pseudo-polynomial
$W(x,y,z)=e^{z}\cdot x+e^{x}\cdot y+e^{y}\cdot z$ can be presented as
$W(x,y,z)=(e^{z}+h(z)y)\cdot x+(e^{x}-h(z)x)\cdot y+e^{y}\cdot z$, where $h$
is arbitrary.

######  Theorem 17.

Let $\mathbf{n}\in\mathbb{N}^{d}$ and let $f\colon\mathbf{I}\to\mathbb{R}$ be
a box-$\mathbf{n}$-convex function. For each $i=1,2,\dots,d$, let
$u_{i1},u_{i2},\dots,u_{in_{i}}$ be pairwise distinct elements of $I_{i}$. If

(10) $f(x_{1},\dots,x_{i-1},u_{ij},x_{i+1},\dots,x_{d})=0$

for each $i=1,2,\dots,d$ and $j=1,2,\dots,n_{i}$, then $f$ is
$\mathbf{n}$-regular.

###### Proof.

Let $A\subset\\{1,2,\dots,d\\}$ and $\mathbf{z}\in\mathbf{I}_{A^{\prime}}$.
Let $a_{1}<a_{2}<\dots<a_{|A|}$ be all elements of the set $A$ and
$a^{\prime}_{1}<a^{\prime}_{2}<\dots<a^{\prime}_{|A^{\prime}|}$ be all
elements of the set $A^{\prime}$. We will show that the function
$f_{A}^{\mathbf{z}}\colon\mathbf{I}_{A}\to\mathbb{R}$ is either
box-$\mathbf{n}_{A}$-convex or box-$\mathbf{n}_{A}$-concave (in particular it
is a linear combination of box-$\mathbf{n}_{A}$-convex functions).

Suppose that there exist $k=1,2,\dots,|A^{\prime}|$, $i=a^{\prime}_{k}$ and
$j=1,2,\dots,n_{i}$ such that $z_{k}=u_{ij}$. Then, by (10),
$f_{A}^{\mathbf{z}}=0$. Consequently, $f_{A}^{\mathbf{z}}$ is both
box-$\mathbf{n}_{A}$-convex and box-$\mathbf{n}_{A}$-concave.

In the remaining case, for each $k=1,2,\dots,|A^{\prime}|$ and
$i=a^{\prime}_{k}$ the numbers $z_{k}$, $u_{i1},u_{i2},\dots,u_{in_{i}}$ are
pairwise distinct elements of $I_{i}$. Let $\mathbf{x}_{i}=(z_{k},u_{i1},$
$u_{i2},\dots$, $u_{in_{i}})$. For each $i\in A$ let
$x_{i0},x_{i1},\dots,x_{in_{i}}$ be pairwise distinct elements of $I_{i}$ and
$\mathbf{x}_{i}=(x_{i0},x_{i1},\dots,x_{in_{i}})$.

By Remark 3,

(11) $\left[\ \LARGE{\begin{subarray}{c}\mathbf{x}_{1}\\\ \\\
\mathbf{x}_{2}\\\ \\\ \dots\\\ \\\
\mathbf{x}_{d}\end{subarray}};f\right]=\left[\
\LARGE{\begin{subarray}{c}\mathbf{x}_{a_{1}}\\\ \\\ \mathbf{x}_{a_{2}}\\\ \\\
\dots\\\ \\\ \mathbf{x}_{a_{|A|}}\end{subarray}};g\right],$

where $g\colon\mathbf{I}_{A}\to\mathbb{R}$ is given by

$g(\mathbf{x})=\left[\LARGE{\begin{subarray}{c}\mathbf{x}_{a^{\prime}_{1}}\\\
\\\ \mathbf{x}_{a^{\prime}_{2}}\\\ \\\ \dots\\\ \\\
\mathbf{x}_{a^{\prime}_{|A^{\prime}|}}\end{subarray}};f_{A^{\prime}}^{\mathbf{x}}\right].$

By (10) and by Proposition 3 applied to the function
$f_{A^{\prime}}^{\mathbf{x}}$ we obtain

$g(\mathbf{x})=\frac{f_{A^{\prime}}^{\mathbf{x}}(z_{1},z_{2},\dots,z_{|A^{\prime}|})}{\prod\limits_{k=1}^{|A^{\prime}|}\prod\limits_{\begin{subarray}{c}j=1\\\
\text{(for
$i=a^{\prime}_{k}$)}\end{subarray}}^{n_{i}}(z_{k}-u_{ij})}=\frac{f_{A}^{\mathbf{z}}(\mathbf{x})}{\prod\limits_{k=1}^{|A^{\prime}|}\prod\limits_{\begin{subarray}{c}j=1\\\
\text{(for $i=a^{\prime}_{k}$)}\end{subarray}}^{n_{i}}(z_{k}-u_{ij})}.$

It follows that (11) can be written as

$\left[\ \LARGE{\begin{subarray}{c}\mathbf{x}_{1}\\\ \\\ \mathbf{x}_{2}\\\ \\\
\dots\\\ \\\
\mathbf{x}_{d}\end{subarray}};f\right]=\frac{1}{\prod\limits_{k=1}^{|A^{\prime}|}\prod\limits_{\begin{subarray}{c}j=1\\\
\text{(for
$i=a^{\prime}_{k}$)}\end{subarray}}^{n_{i}}(z_{k}-u_{ij})}\cdot\left[\
\LARGE{\begin{subarray}{c}\mathbf{x}_{a_{1}}\\\ \\\ \mathbf{x}_{a_{2}}\\\ \\\
\dots\\\ \\\ \mathbf{x}_{a_{|A|}}\end{subarray}};f_{A}^{\mathbf{z}}\right].$

The left side of the above equality in non-negative (because $f$ is
box-$\mathbf{n}$-convex), which implies that the above multiple divided
difference of $f_{A}^{\mathbf{z}}$ has the same sign as the denominator.
Consequently, $f_{A}^{\mathbf{z}}$ is box-$\mathbf{n}_{A}$-convex if the
denominator is positive, and it is box-$\mathbf{n}_{A}$-concave if the
denominator is negative. The proposition is proved. ∎

By Lemma 10 and Theorem 17, we obtain the following proposition.

######  Proposition 6.

Let $\mathbf{n}=(n_{1},n_{2},\dots,n_{d})\in\mathbb{N}^{d}$ and let
$f\colon\mathbf{I}\to\mathbb{R}$ be a box-$\mathbf{n}$-convex function. There
exists a pseudo-polynomial $W\colon\mathbf{I}\to\mathbb{R}$ of degree
$(n_{1}-1,n_{2}-1,\dots,n_{d}-1)$ such that $f-W$ is an $\mathbf{n}$-regular
and box-$\mathbf{n}$-convex function.

######  Lemma 18.

Let $\mathbf{n}\in\mathbb{N}^{d}$. For every $\mathbf{n}$-regular function
$f\colon\mathbf{I}\to\mathbb{R}$ there exist two $\mathbf{n}$-regular and
box-$\mathbf{n}$-convex functions $g,h\colon\mathbf{I}\to\mathbb{R}$ such that
$f=g-h$.

###### Proof.

By Definition 4, $f=\sum_{j=1}^{J}a_{j}f_{j}$, where
$f_{1},f_{2},\dots,f_{J}\colon\mathbf{I}\to\mathbb{R}$ are
box-$\mathbf{n}$-convex functions. We put $\widetilde{g}=\sum_{j\colon
a_{j}\geq 0}a_{j}f_{j}$ and $\widetilde{h}=\sum_{j\colon
a_{j}<0}(-a_{j})f_{j}$. Then $f=\widetilde{g}-\widetilde{h}$ and both
$\widetilde{g}$ and $\widetilde{h}$ are box-$\mathbf{n}$-convex functions. By
Proposition 6, there exists box-$\mathbf{n}$-affine pseudo-polynomial $W$ such
that $\widetilde{g}-W$ is $\mathbf{n}$-regular and box-$\mathbf{n}$-convex.
Let $g=\widetilde{g}-W$ and $h=\widetilde{h}-W$. Then both $g$ and $h$ are
box-$\mathbf{n}$-convex, and $f=g-h$. Since $f$ and $g$ are
$\mathbf{n}$-regular, we obtain that $h=g-f$ is also $\mathbf{n}$-regular. ∎

######  Lemma 19.

Let $\mathbf{n}\in\mathbb{N}^{d}$, $i\in\\{1,2,\dots,d\\}$ and let
$u_{i1},u_{i2},\dots,u_{in_{i}}\in I_{i}$ be pairwise distinct. Assume that
$f\colon\mathbf{I}\to\mathbb{R}$ is an $\mathbf{n}$-regular function such that
$f(x_{1},x_{2},\dots,x_{d})=0$, whenever $x_{i}=u_{ij}$ for
$j=1,2,\dots,n_{i}$. Then there exist two $\mathbf{n}$-regular and
box-$\mathbf{n}$-convex functions $g,h\colon\mathbf{I}\to\mathbb{R}$ such that
$f=g-h$ and $g(x_{1},x_{2},\dots,x_{d})=h(x_{1},x_{2},\dots,x_{d})=0$,
whenever $x_{i}=u_{ij}$ for $j=1,2,\dots,n_{i}$.

###### Proof.

The proof of Lemma 19 is, essentially, the same as the proof of Lemma 18. The
only difference is that, according to Lemma 10, we can choose the pseudo-
polynomial $W$ such that the following additional condition is satisfied:
$W(x_{1},x_{2},\dots,x_{d})=\widetilde{g}(x_{1},x_{2},\dots,x_{d})$, whenever
$x_{i}=u_{ij}$ for $j=1,2,\dots,n_{i}$. ∎

##  6\. Integration and differentiation of $\mathbf{n}$-regular functions

In this section we assume that all intervals $I_{1},I_{2},\dots,I_{d}$ are
open intervals (bounded or unbounded).

######  Lemma 20.

Let $i\in\\{1,2,\dots,d\\}$ be fixed and $\alpha_{i}\in I_{i}$. Let
$\mathbf{n}=(n_{1},n_{2},\dots,n_{d})\in\mathbb{N}^{d}$ be such that
$n_{i}\geq 2$ and let $f\colon\mathbf{I}\to\mathbb{R}$ be an
$\mathbf{n}$-regular function.

Let $\psi\colon\mathbf{I}\to\mathbb{R}$ be right-derivative of $f$ with
respect to the $i$th variable, i.e. $\psi$ is such that for every
$\mathbf{y}\in\mathbf{I}_{\\{i\\}^{\prime}}$ the function
$\psi_{\\{i\\}}^{\mathbf{y}}$ is the right-derivative of
$f_{\\{i\\}}^{\mathbf{y}}$. Then, the function $\psi$ is well defined.
Moreover, $\psi$ is a $(n_{1},\dots,n_{i}-1,\dots,n_{d})$-regular function,
and it satisfies the equation $f(x_{1},\dots,x_{d})=$
$f(x_{1},\dots,\alpha_{i},\dots,x_{d})+\int_{\alpha_{i}}^{x_{i}}\psi(x_{1},\dots,t,\dots,x_{d})dt$,
where $t$ and $\alpha_{i}$ stand at the $i$th position.

If, in addition, $f$ is box-$\mathbf{n}$-convex, then the function $\psi$ is
box-$(n_{1},\dots,n_{i}-1,\dots,n_{d})$-convex.

###### Proof.

For every $\mathbf{y}\in\mathbf{I}_{\\{i\\}^{\prime}}$ the function
$f_{\\{i\\}}^{\mathbf{y}}$ is a linear combination of box-$(n_{i})$-convex
functions. Taking into account Remark 6 and $n_{i}\geq 2$, we obtain that
$\psi$ is well defined and
$f(\mathbf{x})=f_{\\{i\\}}^{\mathbf{y}}(x_{i})=f_{\\{i\\}}^{\mathbf{y}}(\alpha_{i})+\int_{\alpha_{i}}^{x_{i}}\psi_{\\{i\\}}^{\mathbf{y}}(t)dt$
for $\mathbf{y}=\mathbf{x}_{\\{i\\}^{\prime}}$.

First we show that if $f$ is $\mathbf{n}$-regular and box-$\mathbf{n}$-convex,
then $\psi$ is box-$(n_{1},\dots,n_{i}-1,\dots,n_{d})$-convex. Let
$\mathbf{x}_{k}=(x_{k0},x_{k1},\dots,x_{kn_{k}})\in I_{k}^{n_{k}+1}$ for
$k\in\\{1,2,\dots,d\\}\setminus\\{i\\}$, and
$\mathbf{x}_{i}=(x_{i1},x_{i2},\dots,x_{in_{i}})\in I_{i}^{n_{i}}$ be vectors
with pairwise distinct coordinates. Using Remark 3 and Lemma 2, we obtain

$\left[\ \LARGE{\begin{subarray}{c}\mathbf{x}_{1}\\\ \\\ \mathbf{x}_{2}\\\ \\\
\dots\\\ \\\ \mathbf{x}_{d}\end{subarray}};\psi\right]=\left[\
\LARGE{\begin{subarray}{c}\mathbf{x}_{1}\\\ \\\ \dots\\\ \\\
\mathbf{x}_{i-1}\\\ \\\ \mathbf{x}_{i+1}\\\ \\\ \dots\\\ \\\
\mathbf{x}_{d}\end{subarray}};\left[\mathbf{x}_{i};\psi_{\\{i\\}}\right]\right]=\left[\
\LARGE{\begin{subarray}{c}\mathbf{x}_{1}\\\ \\\ \dots\\\ \\\
\mathbf{x}_{i-1}\\\ \\\ \mathbf{x}_{i+1}\\\ \\\ \dots\\\ \\\
\mathbf{x}_{d}\end{subarray}};\left[x_{i1},x_{i2},\ldots,x_{in_{i}};\psi_{\\{i\\}}\right]\right]\\\
=\left[\ \LARGE{\begin{subarray}{c}\mathbf{x}_{1}\\\ \\\ \dots\\\ \\\
\mathbf{x}_{i-1}\\\ \\\ \mathbf{x}_{i+1}\\\ \\\ \dots\\\ \\\
\mathbf{x}_{d}\end{subarray}};\sum_{j=1}^{n_{i}}\lim_{x_{i0}\downarrow
x_{ij}}\left[x_{i0},x_{i1},\ldots,x_{in_{i}};f_{\\{i\\}}\right]\right]=\sum_{j=1}^{n_{i}}\lim_{x_{i0}\downarrow
x_{ij}}\left[\LARGE{\begin{subarray}{c}x_{10},\ x_{11},\ldots,\ x_{1n_{1}}\\\
\\\ x_{20},\ x_{21},\ldots,\ x_{2n_{2}}\\\ \\\ \dots\\\ \\\ x_{d0},\
x_{d1},\ldots,\ x_{dn_{d}}\end{subarray}};f\right]\geq 0$

(the last inequality follows from the box-$\mathbf{n}$-convexity of $f$). We
conclude that $\psi$ is box-$(n_{1},\dots,n_{i}-1,\dots,n_{d})$-convex.

Now we prove that $\psi$ is $(n_{1},\dots,n_{i}-1,\dots,n_{d})$-regular. We
put $\widetilde{\mathbf{n}}=(n_{1},\dots,n_{i}-1,\dots,n_{d})$. Let
$A\subset\\{1,2,\dots,d\\}$ and $\mathbf{z}\in\mathbf{I}_{A^{\prime}}$ be
fixed. We need to show that $\psi_{A}^{\mathbf{z}}$ is a linear combination of
box-$\widetilde{\mathbf{n}}_{A}$-convex functions. Let
$(a_{1},a_{2},\dots,a_{|A|})$ be the ordered sequence of the elements of $A$
(i.e., $A=\\{a_{1},a_{2},\dots,a_{|A|}\\}$ and $a_{1}<a_{2}<\dots<a_{|A|}$).
Similarly, let
$(a^{\prime}_{1},a^{\prime}_{2},\dots,a^{\prime}_{|A^{\prime}|})$ be the
ordered sequence of the elements of $A^{\prime}$. We have two cases: $i\in A$
or $i\notin A$.

First we consider the case $i\in A$. Then there exists $l$ such that
$i=a_{l}$. By Lemma 18, there exist two $\mathbf{n}_{A}$-regular and
box-$\mathbf{n}_{A}$-convex functions $g,h\colon\mathbf{I}_{A}\to\mathbb{R}$
such that $f_{A}^{\mathbf{z}}=g-h$. Let $\xi$ and $\eta$ be right-derivatives
of $g$ and $h$, respectively, with respect to the $l$th variable. By the first
part of the proof we obtain that $\xi$ and $\eta$ are
box-$\widetilde{\mathbf{n}}_{A}$-convex functions. Consequently,
$\psi_{A}^{\mathbf{z}}=\xi-\eta$ is a linear combination of
box-$\widetilde{\mathbf{n}}_{A}$-convex functions.

Now we pass to the case $i\notin A$. In that case
$\widetilde{\mathbf{n}}_{A}=\mathbf{n}_{A}$. Since $i\in A^{\prime}$, there
exists $l$ such that $i=a^{\prime}_{l}$. We see that both $f_{A}^{\mathbf{z}}$
and $\psi_{A}^{\mathbf{z}}$ do not depend on $x_{i}$ (the $i$th variable of
$f$ and $\psi$), which is fixed at $z_{l}$. We fix pairwise distinct
$u_{1},u_{2},\dots,u_{n_{i}}\in I_{i}$ such that $u_{1}=z_{l}$. By Lemma 9,
there exists a pseudo-polynomial $W$ of the form
$W(x_{1},x_{2},\dots,x_{d})=\sum_{k=0}^{n_{i}-1}A_{ik}(\mathbf{x}_{\\{i\\}^{\prime}})x_{i}^{k}$
such that $f(x_{1},x_{2},\dots,x_{d})=W(x_{1},x_{2},\dots,x_{d})$, whenever
$x_{i}=u_{j}$ for $j=1,2,\dots,n_{i}$. Then the coefficients
$A_{i0},A_{i1},\dots,A_{i,n_{i}-1}$ are
$\mathbf{n}_{\\{i\\}^{\prime}}$-regular functions. We denote
$\widetilde{f}=f-W$. Let $\widetilde{\psi}$ be a right-derivative of
$\widetilde{f}$ with respect to the $i$th variable $x_{i}$. Then
$\widetilde{\psi}=\psi-V$, where
$V(\mathbf{x})=\sum_{k=1}^{n_{i}-1}A_{ik}(\mathbf{x}_{\\{i\\}^{\prime}})kx_{i}^{k-1}$.
Obviously, the pseudo-polynomial $V$ is $\widetilde{\mathbf{n}}$-regular.

We consider the function $\widetilde{f}_{A\cup\\{i\\}}^{\
\widetilde{\mathbf{z}}}$, where
$\widetilde{\mathbf{z}}=(z_{1},\dots,z_{l-1},z_{l+1},\dots,z_{|A^{\prime}|})$.
Without loss of generality (changing the order of variables, if necessary), we
have $\widetilde{f}_{A\cup\\{i\\}}^{\ \widetilde{\mathbf{z}}}\colon
I_{i}\times\prod_{k\in A}I_{k}\to\mathbb{R}$. Using Lemma 19 we obtain two
$\mathbf{n}_{A\cup\\{i\\}}$-regular and box-$\mathbf{n}_{A\cup\\{i\\}}$-convex
functions $g,h\colon I_{i}\times\prod_{k\in A}I_{k}\to\mathbb{R}$ such that
$\widetilde{f}_{A\cup\\{i\\}}^{\ \widetilde{\mathbf{z}}}=g-h$ and
$g(v_{0},v_{1},\dots,v_{|A|})=h(v_{0},v_{1},\dots,v_{|A|})=0$, whenever
$v_{0}=u_{j}$ for $j=1,2,\dots,n_{i}$.

Let $\xi$ and $\eta$ be right-derivatives of $g$ and $h$, respectively, with
respect to the variable $v_{0}$. By (4), for $v_{0}=u_{1}=z_{l}$ we obtain

$\xi_{\\{0\\}^{\prime}}^{(u_{1})}(\widetilde{\mathbf{v}})=\xi_{\\{0\\}}^{\
\widetilde{\mathbf{v}}}(u_{1})=\lim_{u_{0}\downarrow
u_{1}}[u_{0},u_{1},u_{2},\dots,u_{n_{i}};g_{\\{0\\}}^{\
\widetilde{\mathbf{v}}}]\cdot\prod_{j=2}^{n_{i}}(u_{1}-u_{j}),$

where $\widetilde{\mathbf{v}}=(v_{1},v_{2},\dots,v_{|A|})$. Analogous identity
can be obtained for $h$ and $\eta$.

For $k=1,2,\dots,|A|$, let $v_{k0},v_{k1},\dots,v_{kn_{a_{k}}}\in I_{a_{k}}$
be pairwise distinct and let
$\mathbf{v}_{k}=(v_{k0},v_{k1},\dots,v_{kn_{a_{k}}})$.

$\left[\ \LARGE{\begin{subarray}{c}\mathbf{v}_{1}\\\ \\\ \mathbf{v}_{2}\\\ \\\
\dots\\\ \\\
\mathbf{v}_{|A|}\end{subarray}};\xi_{\\{0\\}^{\prime}}^{(u_{1})}\right]=\left[\
\LARGE{\begin{subarray}{c}\mathbf{v}_{1}\\\ \\\ \mathbf{v}_{2}\\\ \\\ \dots\\\
\\\ \mathbf{v}_{|A|}\end{subarray}};\lim_{u_{0}\downarrow
u_{1}}[u_{0},u_{1},\dots,u_{n_{i}};g_{\\{0\\}}]\cdot\prod_{j=2}^{n_{i}}(u_{1}-u_{j})\right]\\\
=\prod_{j=2}^{n_{i}}(u_{1}-u_{j})\cdot\lim_{u_{0}\downarrow u_{1}}\left[\
\LARGE{\begin{subarray}{c}\mathbf{v}_{1}\\\ \\\ \mathbf{v}_{2}\\\ \\\ \dots\\\
\\\
\mathbf{v}_{|A|}\end{subarray}};[u_{0},u_{1},\dots,u_{n_{i}};g_{\\{0\\}}]\right]=\prod_{j=2}^{n_{i}}(u_{1}-u_{j})\cdot\lim_{u_{0}\downarrow
u_{1}}\left[\ \LARGE{\begin{subarray}{c}u_{0},\ u_{1},\dots,\ u_{n_{i}}\\\ \\\
v_{10},\ v_{11},\ \dots,\ v_{1n_{a_{1}}}\\\ \\\ \dots\\\ \\\ v_{|A|0},\
v_{|A|1},\ \dots,\ v_{|A|n_{a_{|A|}}}\end{subarray}};g\right],$

which has the same sign as $\prod_{j=2}^{n_{i}}(u_{1}-u_{j})$ (by
box-$\mathbf{n}_{A\cup\\{i\\}}$-convexity of $g$). It follows that
$\xi_{\\{0\\}^{\prime}}^{(u_{1})}$ is either box-$\mathbf{n}_{A}$-convex or
box-$\mathbf{n}_{A}$-concave. Analogously, we obtain that
$\eta_{\\{0\\}^{\prime}}^{(u_{1})}$ is either box-$\mathbf{n}_{A}$-convex or
box-$\mathbf{n}_{A}$-concave. Therefore the function
$\widetilde{\psi}_{A}^{\mathbf{z}}=\xi_{\\{0\\}^{\prime}}^{(u_{1})}-\eta_{\\{0\\}^{\prime}}^{(u_{1})}$
is a linear combination of box-$\mathbf{n}_{A}$-convex functions. Taking into
account $\widetilde{\mathbf{n}}_{A}=\mathbf{n}_{A}$ and the equality
$\psi_{A}^{\mathbf{z}}=\widetilde{\psi}_{A}^{\mathbf{z}}+V_{A}^{\mathbf{z}}$,
we see, that $\psi_{A}^{\mathbf{z}}$ is a linear combination of
box-$\widetilde{\mathbf{n}}_{A}$-convex functions. ∎

######  Lemma 21.

Let $i\in\\{1,2,\dots,d\\}$ be fixed and $\alpha_{i}\in I_{i}$. Let
$\mathbf{n}=(n_{1},n_{2},\dots,n_{d})\in\mathbb{N}^{d}$ be such that
$n_{i}\geq 1$ and let $f\colon\mathbf{I}\to\mathbb{R}$ be an
$\mathbf{n}$-regular function.

Let $F\colon\mathbf{I}\to\mathbb{R}$ be the function given by the formula

$F(x_{1},\dots,x_{d})=\int_{\alpha_{i}}^{x_{i}}f(x_{1},\dots,t,\dots,x_{d})dt,$

where $t$ stands at the $i$th position. Then $F$ is well defined and it is an
$(n_{1},\dots,n_{i}+1,\dots,n_{d})$-regular function.

If, in addition, $f$ is box-$\mathbf{n}$-convex, then the function $F$ is
box-$(n_{1},\dots,n_{i}+1,\dots,n_{d})$-convex.

###### Proof.

For every $\mathbf{y}\in\mathbf{I}_{\\{i\\}^{\prime}}$ the function
$f_{\\{i\\}}^{\mathbf{y}}$ is a linear combination of box-$(n_{i})$-convex
functions. Taking into account Remark 6 and $n_{i}\geq 1$, we obtain that $F$
is well defined.

First we show that if $f$ is $\mathbf{n}$-regular and box-$\mathbf{n}$-convex,
then $F$ is box-$(n_{1},\dots,n_{i}+1,\dots,n_{d})$-convex. Let
$\mathbf{x}_{k}=(x_{k0},x_{k1},\dots,x_{kn_{k}})\in I_{k}^{n_{k}+1}$ for
$k\in\\{1,2,\dots,d\\}\setminus\\{i\\}$, and
$\mathbf{x}_{i}=(x_{i0},x_{i1},\dots,x_{i,n_{i}+1})\in I_{i}^{n_{i}+2}$ be
vectors with pairwise distinct coordinates. Using Remark 3 and Lemma 1, we
obtain

$\left[\ \LARGE{\begin{subarray}{c}\mathbf{x}_{1}\\\ \\\ \mathbf{x}_{2}\\\ \\\
\dots\\\ \\\ \mathbf{x}_{d}\end{subarray}};F\right]=\left[\
\LARGE{\begin{subarray}{c}\mathbf{x}_{1}\\\ \\\ \dots\\\ \\\
\mathbf{x}_{i-1}\\\ \\\ \mathbf{x}_{i+1}\\\ \\\ \dots\\\ \\\
\mathbf{x}_{d}\end{subarray}};\left[\mathbf{x}_{i};F_{\\{i\\}}\right]\right]=\left[\
\LARGE{\begin{subarray}{c}\mathbf{x}_{1}\\\ \\\ \dots\\\ \\\
\mathbf{x}_{i-1}\\\ \\\ \mathbf{x}_{i+1}\\\ \\\ \dots\\\ \\\
\mathbf{x}_{d}\end{subarray}};\left[x_{i0},x_{i1},\ldots,x_{i,n_{i}+1};F_{\\{i\\}}\right]\right]\\\
=\left[\ \LARGE{\begin{subarray}{c}\mathbf{x}_{1}\\\ \\\ \dots\\\ \\\
\mathbf{x}_{i-1}\\\ \\\ \mathbf{x}_{i+1}\\\ \\\ \dots\\\ \\\
\mathbf{x}_{d}\end{subarray}};\int_{0}^{1}t^{n_{i}}\left[y_{1,t},y_{2,t},\ldots,y_{n_{i}+1,t};f_{\\{i\\}}\right]dt\right]=\int_{0}^{1}t^{n_{i}}\left[\LARGE{\begin{subarray}{c}x_{10},\
x_{11},\ldots,\ x_{1n_{1}}\\\ \\\ \dots\\\ \\\ y_{1,t},\ y_{2,t},\ldots,\
y_{n_{i}+1,t}\\\ \\\ \dots\\\ \\\ x_{d0},\ x_{d1},\ldots,\
x_{dn_{d}}\end{subarray}};f\right]dt\geq 0,$

where $y_{j,t}=tx_{ij}+(1-t)x_{i0}$ for $t\in[0,1]$ and $j=1,2,\dots,n_{i}+1$.
The last inequality follows from the box-$\mathbf{n}$-convexity of $f$. We
conclude that $F$ is box-$(n_{1},\dots,n_{i}+1,\dots,n_{d})$-convex.

Now we prove that $F$ is $(n_{1},\dots,n_{i}+1,\dots,n_{d})$-regular. We put
$\widetilde{\mathbf{n}}=(n_{1},\dots,n_{i}+1,\dots,n_{d})$. Let
$A\subset\\{1,2,\dots,d\\}$ and $\mathbf{z}\in\mathbf{I}_{A^{\prime}}$ be
fixed. We need to show that $F_{A}^{\mathbf{z}}$ is a linear combination of
box-$\widetilde{\mathbf{n}}_{A}$-convex functions. Let
$(a_{1},a_{2},\dots,a_{|A|})$ be the ordered sequence of the elements of $A$
(i.e., $A=\\{a_{1},a_{2},\dots,a_{|A|}\\}$ and $a_{1}<a_{2}<\dots<a_{|A|}$).
Similarly, let
$(a^{\prime}_{1},a^{\prime}_{2},\dots,a^{\prime}_{|A^{\prime}|})$ be the
ordered sequence of the elements of $A^{\prime}$. We have two cases: $i\in A$
or $i\notin A$.

First we consider the case $i\in A$. Then there exists $l$ such that
$i=a_{l}$. By Lemma 18, there exist two $\mathbf{n}_{A}$-regular and
box-$\mathbf{n}_{A}$-convex functions $g,h\colon\mathbf{I}_{A}\to\mathbb{R}$
such that $f_{A}^{\mathbf{z}}=g-h$. Let
$G(y_{1},\dots,y_{|A|})=\int_{\alpha_{i}}^{y_{l}}g(y_{1},\dots,t,\dots,y_{|A|})dt$
and
$H(y_{1},\dots,y_{|A|})=\int_{\alpha_{i}}^{y_{l}}h(y_{1},\dots,t,\dots,y_{|A|})dt$,
where $t$ stands at the $l$th position. By the first part of the proof we
obtain that $G$ and $H$ are box-$\widetilde{\mathbf{n}}_{A}$-convex functions.
Consequently, $F_{A}^{\mathbf{z}}=G-H$ is a linear combination of
box-$\widetilde{\mathbf{n}}_{A}$-convex functions.

Now we pass to the case $i\notin A$. In that case
$\widetilde{\mathbf{n}}_{A}=\mathbf{n}_{A}$. Since $i\in A^{\prime}$, there
exists $l$ such that $i=a^{\prime}_{l}$. We see that both $f_{A}^{\mathbf{z}}$
and $F_{A}^{\mathbf{z}}$ do not depend on $x_{i}$ (the $i$th variable of $f$
and $F$), which is fixed at $z_{l}$. We fix pairwise distinct
$u_{1},u_{2},\dots,u_{n_{i}}\in I_{i}$ such that
$u_{j}<\min(\alpha_{i},z_{l})$ for $j=1,2,\dots,n_{i}$. By Lemma 9, there
exists a pseudo-polynomial $W$ of the form
$W(x_{1},x_{2},\dots,x_{d})=\sum_{k=0}^{n_{i}-1}A_{ik}(\mathbf{x}_{\\{i\\}^{\prime}})x_{i}^{k}$
such that $f(x_{1},x_{2},\dots,x_{d})=W(x_{1},x_{2},\dots,x_{d})$, whenever
$x_{i}=u_{j}$ for $j=1,2,\dots,n_{i}$. Then the coefficients
$A_{i0},A_{i1},\dots,A_{i,n_{i}-1}$ are
$\mathbf{n}_{\\{i\\}^{\prime}}$-regular functions. We denote
$\widetilde{f}=f-W$. Let

$\widetilde{F}(x_{1},\dots,x_{d})=\int_{\alpha_{i}}^{x_{i}}\widetilde{f}(x_{1},\dots,t,\dots,x_{d})dt,$

where $t$ stands at the $i$th position. Then $\widetilde{F}=F-V$, where

$V(\mathbf{x})=\sum_{k=0}^{n_{i}-1}A_{ik}(\mathbf{x}_{\\{i\\}^{\prime}})\frac{x_{i}^{k+1}-\alpha_{i}^{k+1}}{k+1}.$

Obviously, the pseudo-polynomial $V$ is $\widetilde{\mathbf{n}}$-regular.

We consider the function $\widetilde{f}_{A\cup\\{i\\}}^{\
\widetilde{\mathbf{z}}}$, where
$\widetilde{\mathbf{z}}=(z_{1},\dots,z_{l-1},z_{l+1},\dots,z_{|A^{\prime}|})$.
Without loss of generality (changing the order of variables, if necessary), we
have $\widetilde{f}_{A\cup\\{i\\}}^{\ \widetilde{\mathbf{z}}}\colon
I_{i}\times\prod_{k\in A}I_{k}\to\mathbb{R}$. Using Lemma 19 we obtain two
$\mathbf{n}_{A\cup\\{i\\}}$-regular and box-$\mathbf{n}_{A\cup\\{i\\}}$-convex
functions $g,h\colon I_{i}\times\prod_{k\in A}I_{k}\to\mathbb{R}$ such that
$\widetilde{f}_{A\cup\\{i\\}}^{\ \widetilde{\mathbf{z}}}=g-h$ and
$g(v_{0},v_{1},\dots,v_{|A|})=h(v_{0},v_{1},\dots,v_{|A|})=0$, whenever
$v_{0}=u_{j}$ for $j=1,2,\dots,n_{i}$. Let
$G(y_{0},y_{1},\dots,y_{|A|})=\int_{\alpha_{i}}^{y_{0}}g(t,y_{1},\dots,y_{|A|})dt$
and
$H(y_{0},y_{1},\dots,y_{|A|})=\int_{\alpha_{i}}^{y_{0}}h(t,y_{1},\dots,y_{|A|})dt$.

For $k=1,2,\dots,|A|$, let $v_{k0},v_{k1},\dots,v_{kn_{a_{k}}}\in I_{a_{k}}$
be pairwise distinct and let
$\mathbf{v}_{k}=(v_{k0},v_{k1},\dots,v_{kn_{a_{k}}})$. By Remark 3 and (3),
for every $t$ between $\alpha_{i}$ and $z_{l}$ we have

$\left[\ \LARGE{\begin{subarray}{c}t,\ u_{1},\dots,\ u_{n_{i}}\\\ \\\ v_{10},\
v_{11},\ \dots,\ v_{1n_{a_{1}}}\\\ \\\ \dots\\\ \\\ v_{|A|0},\ v_{|A|1},\
\dots,\ v_{|A|n_{a_{|A|}}}\end{subarray}};g\right]=\left[\
\LARGE{\begin{subarray}{c}\mathbf{v}_{1}\\\ \\\ \mathbf{v}_{2}\\\ \\\ \dots\\\
\\\
\mathbf{v}_{|A|}\end{subarray}};[t,u_{1},\dots,u_{n_{i}};g_{\\{0\\}}]\right]\\\
=\left[\ \LARGE{\begin{subarray}{c}\mathbf{v}_{1}\\\ \\\ \mathbf{v}_{2}\\\ \\\
\dots\\\ \\\
\mathbf{v}_{|A|}\end{subarray}};\frac{g_{\\{0\\}^{\prime}}^{(t)}}{\prod_{j=1}^{n_{i}}(t-u_{j})}\right]=\frac{1}{\prod_{j=1}^{n_{i}}(t-u_{j})}\left[\
\LARGE{\begin{subarray}{c}\mathbf{v}_{1}\\\ \\\ \mathbf{v}_{2}\\\ \\\ \dots\\\
\\\ \mathbf{v}_{|A|}\end{subarray}};g_{\\{0\\}^{\prime}}^{(t)}\right].$

Since $t-u_{j}>t-\min(\alpha_{i},z_{l})\geq 0$ for each $j$, and $g$ is
box-$\mathbf{n}_{A\cup\\{i\\}}$-convex, we obtain that $\left[\
\LARGE{\begin{subarray}{c}\mathbf{v}_{1}\\\ \\\ \mathbf{v}_{2}\\\ \\\ \dots\\\
\\\ \mathbf{v}_{|A|}\end{subarray}};g_{\\{0\\}^{\prime}}^{(t)}\right]\geq 0$.
It follows that

$\left[\ \LARGE{\begin{subarray}{c}\mathbf{v}_{1}\\\ \\\ \mathbf{v}_{2}\\\ \\\
\dots\\\ \\\
\mathbf{v}_{|A|}\end{subarray}};G_{\\{0\\}^{\prime}}^{(z_{l})}\right]=\left[\
\LARGE{\begin{subarray}{c}\mathbf{v}_{1}\\\ \\\ \mathbf{v}_{2}\\\ \\\ \dots\\\
\\\
\mathbf{v}_{|A|}\end{subarray}};\int_{\alpha_{i}}^{z_{l}}g_{\\{0\\}^{\prime}}^{(t)}dt\right]=\int_{\alpha_{i}}^{z_{l}}\left[\
\LARGE{\begin{subarray}{c}\mathbf{v}_{1}\\\ \\\ \mathbf{v}_{2}\\\ \\\ \dots\\\
\\\ \mathbf{v}_{|A|}\end{subarray}};g_{\\{0\\}^{\prime}}^{(t)}\right]dt$

is either non-negative (if $z_{l}\geq\alpha_{i}$) or non-positive (if
$z_{l}<\alpha_{i}$). Consequently, $G_{\\{0\\}^{\prime}}^{(z_{l})}$ is
box-$\mathbf{n}_{A}$-convex or box-$\mathbf{n}_{A}$-concave. Similarly,
$H_{\\{0\\}^{\prime}}^{(z_{l})}$ is box-$\mathbf{n}_{A}$-convex or
box-$\mathbf{n}_{A}$-concave. Therefore the function
$\widetilde{F}_{A}^{\mathbf{z}}=G_{\\{0\\}^{\prime}}^{(z_{l})}-H_{\\{0\\}^{\prime}}^{(z_{l})}$
is a linear combination of box-$\mathbf{n}_{A}$-convex functions. Taking into
account $\widetilde{\mathbf{n}}_{A}=\mathbf{n}_{A}$ and the equality
$F_{A}^{\mathbf{z}}=\widetilde{F}_{A}^{\mathbf{z}}+V_{A}^{\mathbf{z}}$, we
see, that $F_{A}^{\mathbf{z}}$ is a linear combination of
box-$\widetilde{\mathbf{n}}_{A}$-convex functions. ∎

##  7\. Integral representation of box-monotone functions

In this section, we study box-$(1,1,\dots,1)$-convex functions. We will call
them _box-monotone_ functions. Our aim is to present the integral
representation of box-monotone functions. The general idea is based on the
observation that box-monotone functions behave similarly to cumulative
distribution functions of probability distributions. However, while cumulative
distribution functions are always bounded and right-continuous (or left-
continuous), box-monotone functions, in general, do not have these properties.

We start from some results concerning real functions of one variable.

Let $I\subset\mathbb{R}$ be an open interval and $f\colon I\to\mathbb{R}$ be a
function with locally finite variation (which is equivalent to $f$ being a
difference of two non-decreasing functions). For $x\in I$ we denote one-side
limits of $f$ at $x$ as $f(x-)=\lim_{u\uparrow x}f(u)$ and
$f(x+)=\lim_{u\downarrow x}f(u)$. We say that $f$ is a  _jump function_ if for
every $x,y\in I$, $x<y$, we have

(12) $f(y-)-f(x+)=\sum_{x<t<y}(f(t+)-f(t-)).$

Note that our assumption that the function $f$ has locally finite variation
implies that $f(t+)-f(t-)=0$ for all but countably many $t\in(x,y)$, and
$\sum_{x<t<y}|f(t+)-f(t-)|<\infty$.

In the following lemma, we obtain a decomposition of a function with locally
finite variation, which is a counterpart of the well known Lebesgue
decomposition of right-continuous functions with finite variation.

######  Lemma 22.

Let $I\subset\mathbb{R}$ be an open interval and let $\alpha\in I$ be fixed.
For a function $f\colon I\to\mathbb{R}$ with locally finite variation, there
exist unique functions $f_{L},f_{R},f_{c}\colon I\to\mathbb{R}$ such that
$f_{L}$ and $f_{R}$ are jump functions, $f_{L}(\alpha)=f_{R}(\alpha)=0$,
$f_{L}$ is a left-continuous function, $f_{R}$ is a right-continuous function,
$f_{c}$ is a continuous function, and $f=f_{L}+f_{R}+f_{c}$.

Moreover, if the function $f$ is non-decreasing, then the functions $f_{L}$,
$f_{R}$ and $f_{c}$ are also non-decreasing.

###### Proof.

First we show that whenever the functions $f_{L}$, $f_{R}$ and $f_{c}$
satisfying the conditions given in the lemma exist, then they are unique.
Assume that for some $f$ there are two such triplets: $f_{L}$, $f_{R}$,
$f_{c}$, and $\widetilde{f}_{L}$, $\widetilde{f}_{R}$, $\widetilde{f}_{c}$.
Then the functions $g_{L}=f_{L}-\widetilde{f}_{L}$,
$g_{R}=f_{R}-\widetilde{f}_{R}$ and $g_{c}=f_{c}-\widetilde{f}_{c}$ form a
triplet for the function $g=f-f=0$. It is enough to show that
$g_{L}=g_{R}=g_{c}=0$. Since $g_{L}=g-g_{R}-g_{c}=-g_{R}-g_{c}$, we obtain
that $g_{L}$ is not only left-continuous, but also right-continuous, hence it
is continuous. Let $x,y\in I$, $x<y$. Since $g_{L}$ is a continuous jump
function, we have

$g_{L}(y)-g_{L}(x)=g_{L}(y-)-g_{L}(x+)=\sum_{x<t<y}(g_{L}(t+)-g_{L}(t-))=0.$

Therefore $g_{L}$ is constant. Taking into account the condition
$g_{L}(\alpha)=0$, we get $g_{L}=0$. Similarly, $g_{R}=0$. Consequently,
$g_{c}=g-g_{L}-g_{R}=0$.

We will show that the following functions $f_{L}$, $f_{R}$ and $f_{c}$ satisfy
the conditions given in the lemma.

$\displaystyle f_{L}(x)$ $\displaystyle=\sum_{\alpha\leq
t<x}(f(t+)-f(t))-\sum_{x\leq t<\alpha}(f(t+)-f(t)),$ $\displaystyle f_{R}(x)$
$\displaystyle=\sum_{\alpha<t\leq
x}(f(t)-f(t-))-\sum_{x<t\leq\alpha}(f(t)-f(t-)),$ $\displaystyle f_{c}(x)$
$\displaystyle=f(x)-f_{L}(x)-f_{R}(x).$

The above functions are well defined because the function $f$ has locally
finite variation. Clearly, $f_{L}(\alpha)=f_{R}(\alpha)=0$ and
$f=f_{L}+f_{R}+f_{c}$.

For $x,y\in I$, $x<y$ we have

(13) $f_{L}(y)-f_{L}(x)=\sum_{x\leq t<y}(f(t+)-f(t))\text{ and
}f_{R}(y)-f_{R}(x)=\sum_{x<t\leq y}(f(t)-f(t-)).$

It follows that for every $x\in I$ we have

$f_{L}(x)-f_{L}(x-)=\lim_{u\uparrow x}(f_{L}(x)-f_{L}(u))=\lim_{u\uparrow
x}\sum_{u\leq t<x}(f(t+)-f(t))=0,$ $f_{L}(x+)-f_{L}(x)=\lim_{u\downarrow
x}(f_{L}(u)-f_{L}(x))=\lim_{u\downarrow x}\sum_{x\leq
t<u}(f(t+)-f(t))=f(x+)-f(x).$

Similarly, for every $x\in I$ we have $f_{R}(x+)-f_{R}(x)=0$ and
$f_{R}(x)-f_{R}(x-)=f(x)-f(x-)$. It follows that $f_{L}$ is left-continuous
and $f_{R}$ is right-continuous. Moreover, for every $x\in I$ we obtain

$f_{c}(x+)-f_{c}(x)=(f(x+)-f(x))-(f_{L}(x+)-f_{L}(x))-(f_{R}(x+)-f_{R}(x))=0,$
$f_{c}(x)-f_{c}(x-)=(f(x)-f(x-))-(f_{L}(x)-f_{L}(x-))-(f_{R}(x)-f_{R}(x-))=0,$

hence $f_{c}$ is continuous.

Now we show that $f_{L}$ and $f_{R}$ are jump functions. Let $x,y\in I$,
$x<y$. For every $t\in(x,y)$ we have

$f_{L}(t+)-f_{L}(t-)=f_{L}(t+)-f_{L}(t)=f(t+)-f(t).$

By (13) and left-continuity of $f_{L}$, we obtain

$\begin{split}\sum_{x<t<y}(f_{L}(t+)-f_{L}(t-))=&\sum_{x<t<y}(f(t+)-f(t))=(f_{L}(y)-f_{L}(x))-(f(x+)-f(x))\\\
=&(f_{L}(y)-f_{L}(x))-(f_{L}(x+)-f_{L}(x))=f_{L}(y-)-f_{L}(x+),\end{split}$

hence $f_{L}$ is a jump function. Similarly, we obtain that $f_{R}$ is a jump
function.

Now we assume that $f$ is non-decreasing. By (13), we immediately obtain that
the functions $f_{L}$ and $f_{R}$ are non-decreasing. We will prove that
$f_{c}$ is also non-decreasing. Aiming at the contradiction, we assume that
there exist $x,y\in I$ such that $x<y$ and $f_{c}(x)>f_{c}(y)$. The inequality
$f_{c}(x)>f_{c}(y)$ can be written as

$(f_{L}(y)-f_{L}(x))+(f_{R}(y)-f_{R}(x))>f(y)-f(x).$

By (13), we get

$\sum_{x<t<y}(f(t+)-f(t-))>f(y-)-f(x+).$

Then there exist $x<t_{1}<t_{2}<\dots<t_{n}<y$ such that

$\sum_{k=1}^{n}(f(t_{k}+)-f(t_{k}-))>f(y-)-f(x+),$

and, consequently, there exist $x<u_{0}<t_{1}<u_{1}<t_{2}<\dots<t_{n}<u_{n}<y$
such that

$\sum_{k=1}^{n}(f(u_{k})-f(u_{k-1}))>\sum_{k=1}^{n}(f(t_{k}+)-f(t_{k}-))>f(u_{n})-f(u_{0}).$

We obtained a contradiction, which implies that $f_{c}$ is non-decreasing. ∎

In the sequel we may continue to deal with the decomposition
$f=f_{L}+f_{R}+f_{c}$ given by Lemma 22. However, for simplicity, we will use
the decomposition of $f$ to two functions, given by the following corollary.

######  Corollary 23.

Let $I\subset\mathbb{R}$ be an open interval and let $\alpha\in I$ be fixed.
For a function $f\colon I\to\mathbb{R}$ with locally finite variation, there
exist unique functions $f_{L},f_{r}\colon I\to\mathbb{R}$ such that $f_{L}$ is
a left-continuous jump function and $f_{r}$ is a right-continuous function
such that $f_{L}(\alpha)=0$ and $f=f_{L}+f_{r}$.

Moreover, if the function $f$ is non-decreasing, then the functions $f_{L}$
and $f_{r}$ are also non-decreasing.

###### Proof.

Let $f_{L}$, $f_{R}$ and $f_{c}$ be given by Lemma 22. We put
$f_{r}=f_{R}+f_{c}$. The uniqueness can be proven similarly as in the proof of
Lemma 22. ∎

######  Remark 24.

Note that the operations $\cdot_{L}$, $\cdot_{R}$, $\cdot_{c}$ and $\cdot_{r}$
are linear operators. To see it, one can use the explicit formulas presented
in Corollary 23 and the proof of Lemma 22. Moreover, we have
$(f_{L})_{L}=f_{L}$, $(f_{R})_{R}=f_{R}$, $(f_{c})_{c}=f_{c}$ and
$(f_{r})_{r}=f_{r}$. It follows that all the considered operators are linear
projections in the space of functions with locally finite variation. In
particular, $(f_{L})_{r}=f_{L}-(f_{L})_{L}=0$. Similarly,
$(f_{r})_{L}=f_{r}-(f_{r})_{r}=0$.

We are ready to investigate box-monotone functions of $d$ variables. First we
need to extend the decomposition presented in Corollary 23 to functions of
many variables. If a function $f\colon\mathbf{I}\to\mathbb{R}$ is
$(1,1,\dots,1)$-regular, then for every $i\in\\{1,2,\dots,d\\}$ and
$\mathbf{y}\in\mathbf{I}_{\\{i\\}^{\prime}}$ the function
$f_{\\{i\\}}^{\mathbf{y}}$ is the function of one variable with locally finite
variation. It follows that the following definition is valid.

######  Definition 5.

Let $f\colon\mathbf{I}\to\mathbb{R}$ be a $(1,1,\dots,1)$-regular function.
For $i=1,2,\dots,$ $d$ and $\alpha_{i}\in I_{i}$, we define the functions
$f_{(i,L)},f_{(i,r)}\colon\mathbf{I}\to\mathbb{R}$ as follows: for every
$\mathbf{y}\in\mathbf{I}_{\\{i\\}^{\prime}}$ we put
$(f_{(i,L)})_{\\{i\\}}^{\mathbf{y}}=(f_{\\{i\\}}^{\mathbf{y}})_{L}$ and
$(f_{(i,r)})_{\\{i\\}}^{\mathbf{y}}=(f_{\\{i\\}}^{\mathbf{y}})_{r}$.

######  Remark 25.

Intuitively, in Definition 5 we apply operation $\cdot_{L}$ or $\cdot_{r}$ to
the $i$th variable of the function $f$. Note that $f=f_{(i,L)}+f_{(i,r)}$ and
for every $\mathbf{y}\in\mathbf{I}_{\\{i\\}^{\prime}}$ we have that
$(f_{(i,L)})_{\\{i\\}}^{\mathbf{y}}$ is a left-continuous jump function,
$(f_{(i,r)})_{\\{i\\}}^{\mathbf{y}}$ is a right-continuous function, and
$(f_{(i,L)})_{\\{i\\}}^{\mathbf{y}}(\alpha_{i})=0$. According to Corollary 23,
$f_{(i,L)}$ and $f_{(i,r)}$ are the unique functions satisfying the above
conditions. Moreover, by Remark 24, we have that $\cdot_{(i,L)}$ and
$\cdot_{(i,r)}$ are linear operators satisfying
$(f_{(i,L)})_{(i,L)}=f_{(i,L)}$, $(f_{(i,r)})_{(i,r)}=f_{(i,r)}$ and
$(f_{(i,L)})_{(i,r)}=(f_{(i,r)})_{(i,L)}=0$.

######  Lemma 26.

Let $f\colon\mathbf{I}\to\mathbb{R}$ be a $(1,1,\dots,1)$-regular function.
For every $i\in\\{1,2,\dots,d\\}$ and $\alpha_{i}\in I_{i}$ the functions
$f_{(i,L)},f_{(i,r)}\colon\mathbf{I}\to\mathbb{R}$ are
$(1,1,\dots,1)$-regular.

If, in addition, $f$ is box-monotone, then the functions $f_{(i,L)}$ and
$f_{(i,r)}$ are also box-monotone.

###### Proof.

First, we prove that if $f$ is $(1,1,\dots,1)$-regular and box-monotone, then
the functions $f_{(i,L)}$ and $f_{(i,r)}$ are box-monotone. Let $g$ be any of
the functions $f$, $f_{(i,L)}$ and $f_{(i,r)}$. The function $g$ is box-
monotone, if for every vectors $(x_{10},x_{11})$ $\in I_{1}^{2}$, …,
$(x_{d0},x_{d1})\in I_{d}^{2}$ with pairwise distinct coordinates, we have

$\left[\ \LARGE{\begin{subarray}{c}x_{10},\ x_{11}\\\ \\\ x_{20},\ x_{21}\\\
\\\ \dots\\\ \\\ x_{d0},\
x_{d1}\end{subarray}};g\right]=[x_{i0},x_{i1};\widetilde{g}]\geq 0,$

where the function $\widetilde{g}\colon I_{i}\to\mathbb{R}$ is given by

$\widetilde{g}(x_{i}):=\left[\LARGE{\begin{subarray}{c}x_{10},\ x_{11}\\\ \\\
\dots\\\ \\\ x_{i-1,0},\ x_{i-1,1}\\\ \\\ x_{i+1,0},\ x_{i+1,1}\\\ \\\
\dots\\\ \\\ x_{d0},\
x_{d1}\end{subarray}};g_{\\{i\\}^{\prime}}^{(x_{i})}\right].$

The above inequality implies that the function $g$ is box-monotone if and only
if $\widetilde{g}$ is non-decreasing (for every $(x_{10},x_{11})$, …,
$(x_{i-1,0},x_{i-1,1})$, $(x_{i+1,0},x_{i+1,1})$, …, $(x_{d0},x_{d1})$). By
the expanded form of $\widetilde{g}$ given by Proposition 3, the function
$\widetilde{g}$ is a linear combination of functions of the form
$g_{\\{i\\}}^{\mathbf{y}}$ (with the coefficients and $\mathbf{y}$’s depending
on $(x_{10},x_{11})$, …, $(x_{i-1,0},x_{i-1,1})$, $(x_{i+1,0},x_{i+1,1})$, …,
$(x_{d0},x_{d1})$ only). Thus, by the linearity of $\cdot_{L}$, $\cdot_{r}$,
$\cdot_{(i,L)}$ and $\cdot_{(i,r)}$, we obtain that
$\widetilde{f_{(i,L)}}=(\widetilde{f})_{L}$ and
$\widetilde{f_{(i,r)}}=(\widetilde{f})_{r}$. Using Corollary 23 and the fact
that $\widetilde{f}$ is non-decreasing, we obtain that the functions
$\widetilde{f_{(i,L)}}$ and $\widetilde{f_{(i,r)}}$ are non-decreasing. We end
up with the conclusion that the functions $f_{(i,L)}$ and $f_{(i,r)}$ are box-
monotone.

Now, we prove the $(1,1,\dots,1)$-regularity of $f_{(i,L)}$ and $f_{(i,r)}$.
Since $f_{(i,r)}=f-f_{(i,L)}$, it is enough to show the
$(1,1,\dots,1)$-regularity of $f_{(i,L)}$. Let $A\subset\\{1,2,\dots,d\\}$ and
$\mathbf{z}\in\mathbf{I}_{A^{\prime}}$ be fixed. We need to show that
$(f_{(i,L)})_{A}^{\mathbf{z}}$ is a linear combination of box-monotone
functions. Let $(a_{1},a_{2},\dots,a_{|A|})$ be the ordered sequence of the
elements of $A$ (i.e., $A=\\{a_{1},a_{2},\dots,a_{|A|}\\}$ and
$a_{1}<a_{2}<\dots<a_{|A|}$). Similarly, let
$(a^{\prime}_{1},a^{\prime}_{2},\dots,a^{\prime}_{|A^{\prime}|})$ be the
ordered sequence of the elements of $A^{\prime}$. We have two cases: $i\in A$
or $i\notin A$.

First we consider the case $i\in A$. Then there exists $l$ such that
$i=a_{l}$. By Lemma 18, there exist two $(1,1,\dots,1)$-regular and box-
monotone functions $g,h\colon\mathbf{I}_{A}\to\mathbb{R}$ such that
$f_{A}^{\mathbf{z}}=g-h$. By the first part of the proof, we obtain that
$g_{(l,L)}$ and $h_{(l,L)}$ are box-monotone functions. Consequently,
$(f_{(i,L)})_{A}^{\mathbf{z}}=g_{(l,L)}-h_{(l,L)}$ is a linear combination of
box-monotone functions.

Now we pass to the case $i\notin A$. Since $i\in A^{\prime}$, there exists $l$
such that $i=a^{\prime}_{l}$. We see that both $f_{A}^{\mathbf{z}}$ and
$(f_{(i,L)})_{A}^{\mathbf{z}}$ do not depend on $x_{i}$ (the $i$th variable of
$f$ and $f_{(i,L)}$), which is fixed at $z_{l}$. If $z_{l}=\alpha_{i}$, then
$(f_{(i,L)})_{A}^{\mathbf{z}}=0$ is a box-monotone function. In the sequel, we
assume that $z_{l}\neq\alpha_{i}$.

We consider the function $f_{A\cup\\{i\\}}^{\ \widetilde{\mathbf{z}}}$, where
$\widetilde{\mathbf{z}}=(z_{1},\dots,z_{l-1},z_{l+1},\dots,z_{|A^{\prime}|})$.
Without loss of generality (changing the order of variables, if necessary), we
have $f_{A\cup\\{i\\}}^{\ \widetilde{\mathbf{z}}}\colon I_{i}\times\prod_{k\in
A}I_{k}\to\mathbb{R}$. Using Lemma 19, we obtain two $(1,1,\dots,1)$-regular
and box-monotone functions $g,h\colon I_{i}\times\prod_{k\in
A}I_{k}\to\mathbb{R}$ such that $f_{A\cup\\{i\\}}^{\
\widetilde{\mathbf{z}}}=g-h$ and we have
$g(v_{0},v_{1},\dots,v_{|A|})=h(v_{0},v_{1},\dots,v_{|A|})=0$ whenever
$v_{0}=\alpha_{i}$ (we number the variables of $g$ and $h$ starting from 0).

For $k=1,2,\dots,|A|$, let $v_{k0},v_{k1}\in I_{a_{k}}$ be pairwise distinct.
By the first part of the proof, we obtain that $g_{(0,L)}$ is the box-monotone
function. Consequently,

$\begin{split}0\leq&\left[\ \LARGE{\begin{subarray}{c}z_{l},\ \alpha_{i}\\\
\\\ v_{10},\ v_{11}\\\ \\\ \dots\\\ \\\ v_{|A|0},\
v_{|A|1}\end{subarray}};g_{(0,L)}\right]=\frac{\left[\
\LARGE{\begin{subarray}{c}v_{10},\ v_{11}\\\ \\\ \dots\\\ \\\ v_{|A|0},\
v_{|A|1}\end{subarray}};(g_{(0,L)})_{\\{0\\}^{\prime}}^{(z_{l})}\right]-\left[\
\LARGE{\begin{subarray}{c}v_{10},\ v_{11}\\\ \\\ \dots\\\ \\\ v_{|A|0},\
v_{|A|1}\end{subarray}};(g_{(0,L)})_{\\{0\\}^{\prime}}^{(\alpha_{i})}\right]}{z_{l}-\alpha_{i}}\\\
\\\ =&\frac{1}{z_{l}-\alpha_{i}}\cdot\left[\
\LARGE{\begin{subarray}{c}v_{10},\ v_{11}\\\ \\\ \dots\\\ \\\ v_{|A|0},\
v_{|A|1}\end{subarray}};(g_{(0,L)})_{\\{0\\}^{\prime}}^{(z_{l})}\right].\end{split}$

It follows that the sign of the last divided difference depends only on the
sign of $z_{l}-\alpha_{i}$. Consequently,
$(g_{(0,L)})_{\\{0\\}^{\prime}}^{(z_{l})}$ is either
box-$(1,1,\dots,1)$-convex or box-$(1,1,\dots,1)$-concave. Similarly, we can
obtain that $(h_{(0,L)})_{\\{0\\}^{\prime}}^{(z_{l})}$ is either
box-$(1,1,\dots,1)$-convex or box-$(1,1,\dots,1)$-concave. We end up with the
conclusion that
$(f_{(i,L)})_{A}^{\mathbf{z}}=(g_{(0,L)})_{\\{0\\}^{\prime}}^{(z_{l})}-(h_{(0,L)})_{\\{0\\}^{\prime}}^{(z_{l})}$
is a linear combination of box-$(1,1,\dots,1)$-convex functions. The proof is
finished. ∎

######  Lemma 27.

Let $f\colon\mathbf{I}\to\mathbb{R}$ be a $(1,1,\dots,1)$-regular function and
let
$\boldsymbol{\alpha}=(\alpha_{1},\alpha_{2},\dots,\alpha_{d})\in\mathbf{I}$ be
fixed. If $i,j\in\\{1,2,\dots,d\\}$ and $a,b\in\\{L,r\\}$, then
$f_{(i,a)(j,b)}=f_{(j,b)(i,a)}$.

By $f_{(i,a)(j,b)}$ we mean $(f_{(i,a)})_{(j,b)}$. The same convention applies
in other cases.

###### Proof.

If $i=j$, then the lemma follows from Remark 25.

First we show that the conclusion of the lemma holds for $a=b=L$. It is enough
to prove that for every $\mathbf{y}\in\mathbf{I}_{\\{i,j\\}^{\prime}}$ and for
$\widetilde{f}=f_{\\{i,j\\}}^{\mathbf{y}}$ we have
$\widetilde{f}_{(1,L)(2,L)}=\widetilde{f}_{(2,L)(1,L)}$. By Lemma 18, there
exist two $(1,1)$-regular and box-monotone functions $g,h\colon I_{i}\times
I_{j}\to\mathbb{R}$ such that $\widetilde{f}=g-h$.

Since $g$ is box-monotone, by Lemma 26, we obtain that both functions
$g_{(1,L)(2,L)(1,r)}$ and $g_{(1,L)(2,r)(1,r)}$ are box-monotone. Therefore

$g_{(1,L)(2,L)(1,r)}=(g_{(1,L)(2,L)}+g_{(1,L)(2,r)})_{(1,r)}-g_{(1,L)(2,r)(1,r)}=g_{(1,L)(1,r)}-g_{(1,L)(2,r)(1,r)}=-g_{(1,L)(2,r)(1,r)}$

is a box-$(1,1)$-affine function. Similarly (exchanging all $\cdot_{(1,L)}$
and $\cdot_{(1,r)}$), we obtain that $g_{(1,r)(2,L)(1,L)}$ is
box-$(1,1)$-affine. It follows that

$\begin{split}g_{(2,L)(1,L)}-g_{(1,L)(2,L)}=&(g_{(1,L)}+g_{(1,r)})_{(2,L)(1,L)}-(g_{(1,L)(2,L)(1,L)}+g_{(1,L)(2,L)(1,r)})\\\
=&g_{(1,r)(2,L)(1,L)}-g_{(1,L)(2,L)(1,r)}\end{split}$

is also box-$(1,1)$-affine. On the other hand, we have
$g_{(2,L)(1,L)}(v_{1},v_{2})=0=g_{(1,L)(2,L)}(v_{1},v_{2})$, whenever
$v_{1}=\alpha_{i}$ or $v_{2}=\alpha_{j}$. By Proposition 4, we get
$g_{(2,L)(1,L)}=g_{(1,L)(2,L)}$. Similarly, we obtain
$h_{(2,L)(1,L)}=h_{(1,L)(2,L)}$. Consequently,
$\widetilde{f}_{(2,L)(1,L)}=\widetilde{f}_{(1,L)(2,L)}$ and
$f_{(i,L)(j,L)}=f_{(j,L)(i,L)}$.

We proved one of the four required identities. We use it to prove the other
three:

$f_{(i,L)(j,r)}=f_{(i,L)}-f_{(i,L)(j,L)}=f_{(i,L)}-f_{(j,L)(i,L)}=f_{(j,r)(i,L)},$
$f_{(i,r)(j,L)}=f_{(j,L)}-f_{(i,L)(j,L)}=f_{(j,L)}-f_{(j,L)(i,L)}=f_{(j,L)(i,r)},$
$f_{(i,r)(j,r)}=f_{(j,r)}-f_{(i,L)(j,r)}=f_{(j,r)}-f_{(j,r)(i,L)}=f_{(j,r)(i,r)}.$

This completes the proof of the lemma. ∎

In the following theorem we study a $d$-dimensional counterpart of the
decomposition introduced in Definition 5.

######  Theorem 28.

Let $f\colon\mathbf{I}\to\mathbb{R}$ be a $(1,1,\dots,1)$-regular function and
let
$\boldsymbol{\alpha}=(\alpha_{1},\alpha_{2},\dots,\alpha_{d})\in\mathbf{I}$ be
fixed. There exist unique $(1,1,\dots,1)$-regular functions $f_{\mathbf{b}}$,
$\mathbf{b}=(b_{1},b_{2},\dots,b_{d})\in\\{L,r\\}^{d}$, such that

$f=\sum_{\mathbf{b}\in\\{L,r\\}^{d}}f_{\mathbf{b}},$

and for every $\mathbf{b}\in\\{L,r\\}^{d}$, $i=1,2,\dots,d$ and
$\mathbf{y}\in\mathbf{I}_{\\{i\\}^{\prime}}$, we have:

  * (a)

if $b_{i}=L$, then $(f_{\mathbf{b}})_{\\{i\\}}^{\mathbf{y}}$ is a left-
continuous jump function,
$(f_{\mathbf{b}})_{\\{i\\}}^{\mathbf{y}}(\alpha_{i})=0$,

  * (b)

if $b_{i}=r$, then $(f_{\mathbf{b}})_{\\{i\\}}^{\mathbf{y}}$ is a right-
continuous function.

If, in addition, $f$ is box-monotone, then the functions $f_{\mathbf{b}}$ are
also box-monotone.

###### Proof.

Let $\mathbf{b}=(b_{1},b_{2},\dots,b_{d})\in\\{L,r\\}^{d}$. We consider the
functions $f_{(b_{1},b_{2},\dots,b_{k})}$, $k=1,\ldots,d$, defined inductively
as follows: $f_{(b_{1})}=f_{(1,b_{1})}$ and
$f_{(b_{1},b_{2},\dots,b_{k})}=(f_{(b_{1},b_{2},\dots,b_{k-1})})_{(k,b_{k})}$,
$k=2,\ldots,d$. We will check that
$f_{\mathbf{b}}=f_{(b_{1},b_{2},\dots,b_{d})}$ satisfies the required
conditions.

By Lemma 26, the functions $f_{\mathbf{b}}$ are $(1,1,\dots,1)$-regular. If,
in addition, $f$ is box-monotone, then the functions $f_{\mathbf{b}}$ are also
box-monotone. By Remark 25, we have

$f=\sum_{b_{1}\in\\{L,r\\}}f_{(b_{1})}=\sum_{b_{1},b_{2}\in\\{L,r\\}}f_{(b_{1},b_{2})}=\dots=\sum_{\mathbf{b}\in\\{L,r\\}^{d}}f_{\mathbf{b}}.$

By Lemma 27, and the last part of Remark 25, we also have
$f_{\mathbf{b}}=(f_{\mathbf{b}})_{(i,b_{i})}$. Consequently, conditions (a)
and (b) of the theorem are satisfied.

It remains to prove the uniqueness of the functions $f_{\mathbf{b}}$. We will
show by induction on $k=0,1,\dots,d$, that for every for
$a_{1},a_{2},\dots,a_{k}\in\\{L,r\\}$ the function

(14) $\sum_{\begin{subarray}{c}\mathbf{b}\in\\{L,r\\}^{d}\\\
(b_{1},\dots,b_{k})=(a_{1},\dots,a_{k})\end{subarray}}f_{\mathbf{b}}$

is uniquely determined. For $k=0$ the expression (14) is equal to $f$. In the
induction step (for $k=0,1,\dots,d-1$) we observe that

$\sum_{\begin{subarray}{c}\mathbf{b}\in\\{L,r\\}^{d}\\\
(b_{1},\dots,b_{k})=(a_{1},\dots,a_{k})\end{subarray}}f_{\mathbf{b}}=\sum_{\begin{subarray}{c}\mathbf{b}\in\\{L,r\\}^{d}\\\
(b_{1},\dots,b_{k})=(a_{1},\dots,a_{k})\\\
b_{k+1}=L\end{subarray}}f_{\mathbf{b}}+\sum_{\begin{subarray}{c}\mathbf{b}\in\\{L,r\\}^{d}\\\
(b_{1},\dots,b_{k})=(a_{1},\dots,a_{k})\\\
b_{k+1}=r\end{subarray}}f_{\mathbf{b}}$

is the decomposition of expression (14) to its $\cdot_{(k+1,L)}$ and
$\cdot_{(k+1,r)}$ part. By Remark 25, this decomposition is unique.
Consequently, for $k=d$ we obtain the uniqueness of the functions
$f_{\mathbf{b}}$. ∎

######  Definition 6.

For $x,y\in\mathbb{R}$ and $b\in\\{L,r\\}$, we define the function
$\chi_{x,y}^{b}\colon\mathbb{R}\to\mathbb{R}$ as follows:

$\chi_{x,y}^{L}(u)=\mathbf{1}_{(-\infty,y)}(u)-\mathbf{1}_{(-\infty,x)}(u)=\begin{cases}1&\text{if
}x\leq u<y,\\\ -1&\text{if }y\leq u<x,\\\ 0&\text{otherwise,}\end{cases}$
$\chi_{x,y}^{r}(u)=\mathbf{1}_{(-\infty,y]}(u)-\mathbf{1}_{(-\infty,x]}(u)=\begin{cases}1&\text{if
}x<u\leq y,\\\ -1&\text{if }y<u\leq x,\\\ 0&\text{otherwise.}\end{cases}$

######  Theorem 29.

Let $f\colon\mathbf{I}\to\mathbb{R}$ be a function. Let
$\mathbf{b}=(b_{1},b_{2},\dots,b_{d})\in\\{L,r\\}^{d}$ and
$\boldsymbol{\alpha}=(\alpha_{1},\alpha_{2},\dots,\alpha_{d})\in\mathbf{I}$ be
fixed. Then the following conditions are equivalent:

  * $\bullet$

The function $f$ is a $(1,1,\dots,1)$-regular and box-monotone such that for
$i=1,2,\dots,d$ and $\mathbf{y}\in\mathbf{I}_{\\{i\\}^{\prime}}$, we have:

    * (a)

if $b_{i}=L$, then $f_{\\{i\\}}^{\mathbf{y}}$ is a left-continuous jump
function,

    * (b)

if $b_{i}=r$, then $f_{\\{i\\}}^{\mathbf{y}}$ is a right-continuous function,

    * (c)

$f_{\\{i\\}}^{\mathbf{y}}(\alpha_{i})=0$.

  * $\bullet$

There exists a Borel measure $\mu$ on $\mathbf{I}$ such that

    * (i)

$f(x_{1},\dots,x_{d})=\idotsint\limits_{\mathbf{I}}\prod_{j=1}^{d}\chi_{\alpha_{j},x_{j}}^{b_{j}}(u_{j})\
d\mu(u_{1},\dots,u_{d}),$

    * (ii)

$\mu(K)<\infty$ for every compact set $K\subset\mathbf{I}$,

    * (iii)

for each $i=1,2,\dots,d$, if $b_{i}=L$, then the marginal measure of $\mu$,
corresponding to the $i$th coordinate is a discrete measure.

Moreover, the measure $\mu$ that appears in representation (i) is unique.

###### Proof.

First we prove the implication ($\Leftarrow$).

By (i) and (ii), we obtain that $f(x_{1},\dots,x_{d})$ is finite for every
$(x_{1},\dots,x_{d})\in\mathbf{I}$. Moreover, $f(x_{1},\dots,x_{d})=0$,
whenever $x_{j}=\alpha_{j}$ for some $j$. Therefore condition (c) is
satisfied.

For $y,z\in\mathbb{R}$, $y<z$, and $b\in\\{L,r\\}$ we define

(15) $J^{b}_{y,z}=\begin{cases}[y,z)&\text{if }b=L,\\\ (y,z]&\text{if
}b=r.\end{cases}$

For $j=1,2,\dots,d$ we take arbitrary points $y_{j},z_{j}\in I_{j}$,
$y_{j}<z_{j}$. Then, we have

$\begin{split}\mu\left(\prod_{j=1}^{d}J^{b_{j}}_{y_{j},z_{j}}\right)=&\idotsint\limits_{\mathbf{I}}\prod_{j=1}^{d}\chi_{y_{j},z_{j}}^{b_{j}}(u_{j})\
d\mu(u_{1},\dots,u_{d})\\\
=&\idotsint\limits_{\mathbf{I}}\prod_{j=1}^{d}(\chi_{\alpha_{j},z_{j}}^{b_{j}}(u_{j})-\chi_{\alpha_{j},y_{j}}^{b_{j}}(u_{j}))\
d\mu(u_{1},\dots,u_{d})\\\
=&\sum_{B\subset\\{1,\dots,d\\}}(-1)^{|B|}\idotsint\limits_{\mathbf{I}}\prod_{j=1}^{d}\chi_{\alpha_{j},\hat{y}_{B,j}}^{b_{j}}(u_{j})\
d\mu(u_{1},\dots,u_{d})\\\
=&\sum_{B\subset\\{1,\dots,d\\}}(-1)^{|B|}f(\hat{y}_{B,1},\dots,\hat{y}_{B,d}),\end{split}$

where $\hat{y}_{B,j}=y_{j}$ for $j\in B$, and $\hat{y}_{B,j}=z_{j}$ for
$j\notin B$. On the other hand, by (5) for $n_{1}=\dots=n_{d}=1$, we obtain

$\left[\ \LARGE{\begin{subarray}{c}y_{1},\ z_{1}\\\ \\\ y_{2},\ z_{2}\\\ \\\
\dots\\\ \\\ y_{d},\
z_{d}\end{subarray}};f\right]=\sum_{B\subset\\{1,\dots,d\\}}\frac{f(\hat{y}_{B,1},\dots,\hat{y}_{B,d})}{\prod_{i=j}^{d}(\hat{y}_{B,j}-\hat{y}_{B^{\prime},j})}=\frac{1}{\prod_{j=1}^{d}(z_{j}-y_{j})}\cdot\sum_{B\subset\\{1,\dots,d\\}}(-1)^{|B|}f(\hat{y}_{B,1},\dots,\hat{y}_{B,d}).$

Consequently,

(16)
$\mu\left(\prod_{j=1}^{d}J^{b_{j}}_{y_{j},z_{j}}\right)=\sum_{B\subset\\{1,\dots,d\\}}(-1)^{|B|}f(\hat{y}_{B,1},\dots,\hat{y}_{B,d})=\prod_{j=1}^{d}(z_{j}-y_{j})\cdot\left[\
\LARGE{\begin{subarray}{c}y_{1},\ z_{1}\\\ \\\ y_{2},\ z_{2}\\\ \\\ \dots\\\
\\\ y_{d},\ z_{d}\end{subarray}};f\right].$

It follows that the divided difference that appears in (16) is non-negative.
Since $y_{j}<z_{j}$ ($j=1,2,\dots,d$) were chosen arbitrarily, the function
$f$ is box-monotone.

By the box-monotonicity of $f$, condition (c) and Theorem 17, we obtain that
$f$ is $(1,1,\dots,1)$-regular.

Now, we will show that condition (b) is satisfied. Let $i=1,2,\dots,d$ be
fixed. Assume that $b_{i}=r$. The functions
$g_{x_{1},\dots,x_{d}}\colon\mathbf{I}\to\mathbb{R}$ given by formula
$g_{x_{1},\dots,x_{d}}(u_{1},\dots,u_{d})=\prod_{j=1}^{d}\chi_{\alpha_{j},x_{j}}^{b_{j}}(u_{j})$
are obviously bounded. Observe, that if $x_{i}\downarrow\hat{x}_{i}$, then
$\chi_{\alpha_{i},x_{i}}^{r}$ is pointwise convergent to
$\chi_{\alpha_{i},\hat{x}_{i}}^{r}$. Thus, $g_{x_{1},\dots,x_{i},\dots,x_{d}}$
converges to $g_{x_{1},\dots,\hat{x}_{i},\dots,x_{d}}$ pointwise.
Consequently, by Lebesgue’s dominated convergence theorem, we get

$\lim_{x_{i}\downarrow\hat{x}_{i}}f(x_{1},\dots,x_{i},\dots,x_{d})=f(x_{1},\dots,\hat{x}_{i},\dots,x_{d}),$

therefore condition (b) holds.

Similarly, one can prove that if $b_{i}=L$, then $f_{\\{i\\}}^{\mathbf{y}}$ is
a left-continuous function for every
$\mathbf{y}\in\mathbf{I}_{\\{i\\}^{\prime}}$. We will show that
$f_{\\{i\\}}^{\mathbf{y}}$ is a jump function. We consider only the case when
$y_{j}\geq\alpha_{j}$ for all $j\neq i$ (all the other cases are similar). Let
$x<y$, $x,y\in I_{i}$. By condition (iii), we have

$\begin{split}f(y-)-f(x+)=&\mu\left(\prod_{j=1}^{i-1}J^{b_{j}}_{\alpha_{j},x_{j}}\times(x,y)\times\prod_{j=i+1}^{d}J^{b_{j}}_{\alpha_{j},x_{j}}\right)=\sum_{x<t<y}\mu\left(\prod_{j=1}^{i-1}J^{b_{j}}_{\alpha_{j},x_{j}}\times\\{t\\}\times\prod_{j=i+1}^{d}J^{b_{j}}_{\alpha_{j},x_{j}}\right)\\\
=&\sum_{x<t<y}(f(t+)-f(t-)).\end{split}$

We obtained that equation (12) holds for each $x<y$, which implies that
$f_{\\{i\\}}^{\mathbf{y}}$ is a jump function. This completes the proof of
condition (a).

Now, we are going to the proof of the implication ($\Rightarrow$).

We need to define the measure $\mu$. First, by formula (16), we define $\mu$
for the rectangles of the form $\prod_{j=1}^{d}J^{b_{j}}_{y_{j},z_{j}}$, where
$y_{j},z_{j}\in I_{j}$, $y_{j}<z_{j}$, and $J^{b}_{y,z}$ is given by (15).
Using the standard methods, we may uniquely extend $\mu$ to a Borel measure on
$\mathbf{I}$. In this step, the assumptions on one-sided continuity of $f$
((a) and (b)) play an important role. We omit the details of the proof of the
extension. Note, that every measure $\mu$ satisfying condition (i) needs to
fulfill equality (16) for all rectangles. Thus, $\mu$ is uniquely determined.

If $K\subset\mathbf{I}$ is compact, then it is a subset of some rectangle
$\mathbf{J}=\prod_{j=1}^{d}J^{b_{j}}_{y_{j},z_{j}}$. Therefore,
$\mu(K)\leq\mu(\mathbf{J})<\infty$, which proves (ii).

We will show (i). Let $(x_{1},x_{2},\dots,x_{d})\in\mathbf{I}$. For each
$j=1,2,\dots,d$ we put
$(y_{j},z_{j})=(\min(x_{j},\alpha_{j}),\max(x_{j},\alpha_{j}))$. Note that if
$y_{j}=\alpha_{j}$, then
$\chi_{\alpha_{j},x_{j}}^{b_{j}}=\chi_{y_{j},z_{j}}^{b_{j}}$. Otherwise, we
have $\chi_{\alpha_{j},x_{j}}^{b_{j}}=-\chi_{y_{j},z_{j}}^{b_{j}}$.
Consequently, letting $k=|\\{j\colon x_{j}<\alpha_{j}\\}|$, we obtain

$\begin{split}\idotsint\limits_{\mathbf{I}}&\prod_{j=1}^{d}\chi_{\alpha_{j},x_{j}}^{b_{j}}(u_{j})\
d\mu(u_{1},\dots,u_{d})=(-1)^{k}\idotsint\limits_{\mathbf{I}}\prod_{j=1}^{d}\chi_{y_{j},z_{j}}^{b_{j}}(u_{j})\
d\mu(u_{1},\dots,u_{d})\\\
=&(-1)^{k}\mu\left(\prod_{j=1}^{d}J^{b_{j}}_{y_{j},z_{j}}\right)=(-1)^{k}\sum_{B\subset\\{1,\dots,d\\}}(-1)^{|B|}f(\hat{y}_{B,1},\dots,\hat{y}_{B,d}).\end{split}$

Using (c), we obtain that the only non-zero summand in the last sum is the one
for $B=\\{j\colon x_{j}<\alpha_{j}\\}$. This summand equals
$(-1)^{|B|}f(\hat{y}_{B,1},\dots,\hat{y}_{B,d})=(-1)^{k}f(x_{1},x_{2},\dots,x_{d})$.
Therefore, we obtain (i).

It remains to show (iii). We fix $i=1,2,\dots,d$ such that $b_{i}=L$. By the
definition of $\mu$ and formula (16), condition (a) implies that the function
$z_{i}\mapsto\mu\left(\prod_{j=1}^{d}J^{b_{j}}_{y_{j},z_{j}}\right)$ is a jump
function. Thus, we have that the $i$th marginal of $\mu$ restricted to any
rectangle of the form $\prod_{j=1}^{d}J^{b_{j}}_{y_{j},z_{j}}$ is a discrete
measure. Consequently, condition (iii) is satisfied. ∎

######  Theorem 30.

Let $f\colon\mathbf{I}\to\mathbb{R}$ be a function. Let
$\boldsymbol{\alpha}=(\alpha_{1},\alpha_{2},\dots,\alpha_{d})\in\mathbf{I}$ be
fixed. Then the following conditions are equivalent:

  * $\bullet$

$f$ is box-monotone,

  * $\bullet$

there exist a pseudo-polynomial $W\colon\mathbf{I}\to\mathbb{R}$ of the degree
$(0,0,\dots,0)$ and Borel measures $\mu_{\mathbf{b}}$ on $\mathbf{I}$
$(\mathbf{b}\in\\{L,r\\}^{d})$ such that

    * (i)

(17)
$f(x_{1},\dots,x_{d})=W(x_{1},\dots,x_{d})+\sum_{\mathbf{b}\in\\{L,r\\}^{d}}\idotsint\limits_{\mathbf{I}}\prod_{j=1}^{d}\chi_{\alpha_{j},x_{j}}^{b_{j}}(u_{j})\
d\mu_{\mathbf{b}}(u_{1},\dots,u_{d}),$

    * (ii)

$\mu_{\mathbf{b}}(K)<\infty$ for every compact set $K\subset\mathbf{I}$ and
$\mathbf{b}\in\\{L,r\\}^{d}$,

    * (iii)

for each $\mathbf{b}\in\\{L,r\\}^{d}$ and $i=1,2,\dots,d$, if $b_{i}=L$, then
the marginal measure of $\mu_{\mathbf{b}}$, corresponding to the $i$th
coordinate is a discrete measure.

Moreover, for a box-monotone function $f$, the pseudopolynomial $W$ and the
measures $\mu_{\mathbf{b}}$ in the representation (17) are uniquely
determined.

###### Proof.

If the function $f$ is of the form (17), then, by Theorems 12 and 29, it is
box-monotone as a sum of box-monotone functions. The proof of the implication
($\Leftarrow$) is completed.

The implication ($\Rightarrow$) is an immediate consequence of Lemma 10 and
Theorems 17, 28 and 29.

Now, we will prove the uniqueness of the representation. Assume that (17)
holds both for $W$, $\mu_{\mathbf{b}}$ and $\hat{W}$, $\hat{\mu}_{\mathbf{b}}$
($\mathbf{b}\in\\{L,r\\}^{d}$). Then we have

$(\hat{W}-W)(x_{1},\dots,x_{d})=\sum_{\mathbf{b}\in\\{L,r\\}^{d}}\idotsint\limits_{\mathbf{I}}\prod_{j=1}^{d}\chi_{\alpha_{j},x_{j}}^{b_{j}}(u_{j})\
d(\mu_{\mathbf{b}}-\hat{\mu}_{\mathbf{b}})(u_{1},\dots,u_{d}).$

We obtain that $\hat{W}-W$ is a pseudo-polynomial of the degree
$(0,0,\dots,0)$ such that $(\hat{W}-W)(x_{1},\dots,x_{d})=0$, whenever
$x_{i}=\alpha_{i}$ for some $i=1,2,\dots,d$. By Proposition 4, it follows that
$W=\hat{W}$. Consequently, by Theorem 28, all the summands on the right side
are zero functions, and by Theorem 29,
$\mu_{\mathbf{b}}=\hat{\mu}_{\mathbf{b}}$ for all
$\mathbf{b}\in\\{L,r\\}^{d}$. This completes the proof of the theorem. ∎

######  Remark 31.

Note, that each $\mathbf{b}\in\\{L,r\\}^{d}$ the summand corresponding to
$\mathbf{b}$ in the representation (17) can be regarded as a generalized
cumulative distribution function of the measure $\mu_{\mathbf{b}}$.

Let us consider the case, when all the measures $\mu_{\mathbf{b}}$ are finite,
which is equivalent to the existence of $M\in\mathbb{R}$ such that for
arbitrary points $y_{j},z_{j}\in I_{j}$, $j=1,2,\dots,d$ we have

(18) $\prod_{j=1}^{d}(z_{j}-y_{j})\cdot\left[\
\LARGE{\begin{subarray}{c}y_{1},\ z_{1}\\\ \\\ y_{2},\ z_{2}\\\ \\\ \dots\\\
\\\ y_{d},\
z_{d}\end{subarray}};f\right]=\sum_{B\subset\\{1,\dots,d\\}}(-1)^{|B|}f(\hat{y}_{B,1},\dots,\hat{y}_{B,d})<M,$

where $\hat{y}_{B,j}=y_{j}$ for $j\in B$, and $\hat{y}_{B,j}=z_{j}$ for
$j\notin B$. In that case formula (17) can be written using probabilistic
tools.

Let $\mathbf{b}\in\\{L,r\\}^{d}$ and let
$c_{\mathbf{b}}=\mu_{\mathbf{b}}(\mathbf{I})<\infty$. There exists a random
vector $\mathbf{X}_{\mathbf{b}}=(X_{\mathbf{b},1},\dots,X_{\mathbf{b},d})$
such that $\mu_{\mathbf{b}}/c_{\mathbf{b}}$ is the probability distribution of
the vector $\mathbf{X}_{\mathbf{b}}$, whenever $c_{\mathbf{b}}>0$. We have

(19)
$\begin{split}\idotsint\limits_{\mathbf{I}}&\prod_{j=1}^{d}\chi_{\alpha_{j},x_{j}}^{b_{j}}(u_{j})\
d\mu_{\mathbf{b}}(u_{1},\dots,u_{d})=c_{\mathbf{b}}\idotsint\limits_{\mathbf{I}}\prod_{j=1}^{d}(\mathbf{1}_{(-\infty,x_{j}\rangle}(u_{j})-\mathbf{1}_{(-\infty,\alpha_{j}\rangle}(u_{j}))\
dP_{\mathbf{X}_{\mathbf{b}}}(u_{1},\dots,u_{d})\\\
=&c_{\mathbf{b}}\sum_{B\subset\\{1,\dots,d\\}}(-1)^{|B|}P(X_{\mathbf{b},1}<_{b_{1}}\hat{x}_{B,1},\dots,X_{\mathbf{b},d}<_{b_{d}}\hat{x}_{B,d}),\end{split}$

where the bracket $\rangle$ in the $j$th factor is either $)$ if $b_{j}=L$ or
$]$ if $b_{j}=r$, the symbol $<_{L}$ denotes $<$, the symbol $<_{r}$ denotes
$\leq$; $\hat{x}_{B,j}=\alpha_{j}$ for $j\in B$, and $\hat{x}_{B,j}=x_{j}$ for
$j\notin B$.

If $B\neq\emptyset$, then there exists $i\in B$, and the corresponding summand
in (19) does not depend on $x_{i}$. Consequently, this summand is a
box-$(1,\dots,1)$-affine function. It follows, that the expression (19) is a
sum of the pseudo-polynomial of degree $(0,\dots,0)$ and the summand
corresponding to $B=\emptyset$, namely ${c_{\mathbf{b}}\cdot
P}(X_{\mathbf{b},1}<_{b_{1}}x_{1},\dots,X_{\mathbf{b},d}<_{b_{d}}x_{d})$.

Finally we obtain that every box-monotone function $f$ satisfying condition
(18) (e.g. every bounded box-monotone function $f$) has the following
probabilistic representation

(20)
$f(x_{1},\dots,x_{d})=V(x_{1},\dots,x_{d})+\sum_{\mathbf{b}\in\\{L,r\\}^{d}}c_{\mathbf{b}}\cdot
P(X_{\mathbf{b},1}<_{b_{1}}x_{1},\dots,X_{\mathbf{b},d}<_{b_{d}}x_{d}),$

where $V$ is a pseudo-polynomial of degree $(0,\dots,0)$, and for every
$\mathbf{b}\in\\{L,r\\}^{d}$
$\mathbf{X}_{\mathbf{b}}=(X_{\mathbf{b},1},\dots,X_{\mathbf{b},d})$ is a
random vector and $c_{\mathbf{b}}\geq 0$. On the other hand, it can be shown
that every function $f$ of the form (20) is box-monotone and it satisfies
condition (18).

##  8\. Integral representation of box-n-convex functions

######  Theorem 32.

Let $I_{1},I_{2},\dots,I_{d}$ be open intervals (bounded or unbounded) and
$f\colon\mathbf{I}\to\mathbb{R}$ be a function. Let
$\boldsymbol{\alpha}=(\alpha_{1},\alpha_{2},\dots,\alpha_{d})\in\mathbf{I}$
and $\mathbf{n}=(n_{1},n_{2},\dots,n_{d})\in\mathbb{N}^{d}$ be such that
$n_{i}\geq 1$ for $i=1,2,\dots,d$.

Then $f$ is box-$\mathbf{n}$-convex, if and only if it is of the form

(21)
$f(x_{1},\dots,x_{d})=W(x_{1},\dots,x_{d})+\sum_{\mathbf{b}\in\\{L,r\\}^{d}}\idotsint\limits_{\mathbf{I}}\prod_{j=1}^{d}\frac{(x_{j}-u_{j})^{n_{j}-1}}{(n_{j}-1)!}\chi_{\alpha_{j},x_{j}}^{b_{j}}(u_{j})\
d\mu_{\mathbf{b}}(u_{1},\dots,u_{d}),$

where $W\colon\mathbf{I}\to\mathbb{R}$ is a pseudo-polynomial of degree
$(n_{1}-1,\dots,n_{d}-1)$ and measures $\mu_{\mathbf{b}}$
$(\mathbf{b}\in\\{L,r\\}^{d})$ satisfy conditions (ii) and (iii) of Theorem 30
$($here we use the standard convention that $0^{0}=1)$.

###### Proof.

We show the implication ($\Rightarrow$). Taking into account Proposition 6, it
is enough to prove the implication in the case, when the function $f$ is
$\mathbf{n}$-regular. We use an induction on $n_{1}+\dots+n_{d}$. If
$n_{1}+\dots+n_{d}=d$, then (21) is a consequence of Theorem 30. If
$n_{1}+\dots+n_{d}>d$, then $n_{i}\geq 2$ for some $i$. By Lemma 20, we have
$f(x_{1},\dots,x_{d})=f(x_{1},\dots,\alpha_{i},\dots,x_{d})+\int_{\alpha_{i}}^{x_{i}}\psi(x_{1},\dots,t,\dots,x_{d})dt$,
where $t$ and $\alpha_{i}$ stand at the $i$th position, and
$\psi\colon\mathbf{I}\to\mathbb{R}$ is a
$(n_{1},\dots,n_{i}-1,\dots,n_{d})$-regular and
box-$(n_{1},\dots,n_{i}-1,\dots,n_{d})$-convex function. By the induction
hypothesis, we have

$\psi(x_{1},\dots,x_{d})=V(x_{1},\dots,x_{d})\\\
+\sum_{\mathbf{b}\in\\{L,r\\}^{d}}\idotsint\limits_{\mathbf{I}}\frac{(x_{i}-u_{i})^{n_{i}-2}}{(n_{i}-2)!}\chi_{\alpha_{i},x_{i}}^{b_{i}}(u_{i})\prod_{j\neq
i}\frac{(x_{j}-u_{j})^{n_{j}-1}}{(n_{j}-1)!}\chi_{\alpha_{j},x_{j}}^{b_{j}}(u_{j})\
d\mu_{\mathbf{b}}(u_{1},\dots,u_{d}).$

Taking
$W(x_{1},\dots,x_{d})=f(x_{1},\dots,\alpha_{i},\dots,x_{d})+\int_{\alpha_{i}}^{x_{i}}V(x_{1},\dots,t,\dots,x_{d})dt$
and observing that

$\int_{\alpha_{i}}^{x_{i}}\frac{(t-u_{i})^{n_{i}-2}}{(n_{i}-2)!}\chi_{\alpha_{i},t}^{b_{i}}(u_{i})dt=\frac{(x_{i}-u_{i})^{n_{i}-1}}{(n_{i}-1)!}\chi_{\alpha_{i},x_{i}}^{b_{i}}(u_{i}),$

we obtain (21).

We pass to the proof of the implication ($\Leftarrow$). We assume that the
function $f$ is of the form (21). It is enough to prove that each of the
summands is box-$\mathbf{n}$-convex. We will proceed by induction on
$n_{1}+\dots+n_{d}$. If $n_{1}+\dots+n_{d}=d$, then the
box-$\mathbf{n}$-convexity of each summand is a consequence of Theorem 30. If
$n_{1}+\dots+n_{d}>d$, then $n_{i}\geq 2$ for some $i$. The
box-$\mathbf{n}$-convexity of

$\idotsint\limits_{\mathbf{I}}\prod_{j=1}^{d}\frac{(x_{j}-u_{j})^{n_{j}-1}}{(n_{j}-1)!}\chi_{\alpha_{j},x_{j}}^{b_{j}}(u_{j})\
d\mu_{\mathbf{b}}(u_{1},\dots,u_{d})\\\ \\\
=\int_{\alpha_{i}}^{x_{i}}\idotsint\limits_{\mathbf{I}}\frac{(t-u_{i})^{n_{i}-2}}{(n_{i}-2)!}\chi_{\alpha_{i},t}^{b_{i}}(u_{i})\prod_{j\neq
i}\frac{(x_{j}-u_{j})^{n_{j}-1}}{(n_{j}-1)!}\chi_{\alpha_{j},x_{j}}^{b_{j}}(u_{j})\
d\mu_{\mathbf{b}}(u_{1},\dots,u_{d})dt$

is a consequence of the induction hypothesis and Lemma 21. The theorem is
proved. ∎

The above integral representation becomes much simpler when all $n_{i}$’s are
greater than $1$.

######  Theorem 33.

Let $I_{1},I_{2},\dots,I_{d}$ be open intervals (bounded or unbounded) and
$f\colon\mathbf{I}\to\mathbb{R}$ be a function. Let
$\boldsymbol{\alpha}=(\alpha_{1},\alpha_{2},\dots,\alpha_{d})\in\mathbf{I}$
and $\mathbf{n}=(n_{1},n_{2},\dots,n_{d})\in\mathbb{N}^{d}$ be such that
$n_{i}\geq 2$ for $i=1,2,\dots,d$.

Then $f$ is box-$\mathbf{n}$-convex, if and only if it is of the form

(22)
$f(x_{1},\dots,x_{d})=W(x_{1},\dots,x_{d})+\idotsint\limits_{\mathbf{I}}\prod_{j=1}^{d}\frac{(x_{j}-u_{j})^{n_{j}-1}}{(n_{j}-1)!}\chi_{\alpha_{j},x_{j}}^{r}(u_{j})\
d\mu(u_{1},\dots,u_{d}),$

where $W\colon\mathbf{I}\to\mathbb{R}$ is a pseudo-polynomial of degree
$(n_{1}-1,\dots,n_{d}-1)$ and the measure $\mu$ is finite on compact sets.

###### Proof.

The implication ($\Leftarrow$) follows from Theorem 32.

We prove the implication $(\Rightarrow$). By Theorem 32, the function $f$ is
of the form (21). Let $\mathbf{b}\in\\{L,r\\}^{d}$. Let $\mathbf{1}_{b=L}=1$
if $b=L$, and $\mathbf{1}_{b=L}=0$ if $b=r$. Since $n_{1},\dots,n_{d}\geq 2$,
we have

$\frac{(x_{j}-u_{j})^{n_{j}-1}}{(n_{j}-1)!}\chi_{\alpha_{j},x_{j}}^{b_{j}}(u_{j})=\frac{(x_{j}-u_{j})^{n_{j}-1}}{(n_{j}-1)!}\chi_{\alpha_{j},x_{j}}^{r}(u_{j})+\frac{(x_{j}-u_{j})^{n_{j}-1}}{(n_{j}-1)!}\mathbf{1}_{\\{\alpha_{j}\\}}(u_{j})\cdot\mathbf{1}_{b_{j}=L}.$

Using the above identity we obtain

(23)
$\begin{split}\idotsint\limits_{\mathbf{I}}&\prod_{j=1}^{d}\frac{(x_{j}-u_{j})^{n_{j}-1}}{(n_{j}-1)!}\chi_{\alpha_{j},x_{j}}^{b_{j}}(u_{j})\
d\mu_{\mathbf{b}}(u_{1},\dots,u_{d})\\\
=&\sum_{B\subset\\{1,\dots,d\\}}\idotsint\limits_{\mathbf{I}}\prod_{j=1}^{d}\frac{(x_{j}-u_{j})^{n_{j}-1}}{(n_{j}-1)!}\hat{\chi}_{B,j}(u_{j})\
d\mu_{\mathbf{b}}(u_{1},\dots,u_{d}),\end{split}$

where
$\hat{\chi}_{B,j}(u_{j})=\mathbf{1}_{\\{\alpha_{j}\\}}(u_{j})\cdot\mathbf{1}_{b_{j}=L}$
for $j\in B$, and $\hat{\chi}_{B,j}(u_{j})=\chi_{\alpha_{j},x_{j}}^{r}(u_{j})$
for $j\notin B$.

If $B\neq\emptyset$, then there exists $i\in B$, and the corresponding summand
in (23) can be written as

$\begin{split}&\idotsint\limits_{\mathbf{I}}\prod_{j=1}^{d}\frac{(x_{j}-u_{j})^{n_{j}-1}}{(n_{j}-1)!}\hat{\chi}_{B,j}(u_{j})\
d\mu_{\mathbf{b}}(u_{1},\dots,u_{d})\\\
=\frac{(x_{i}-\alpha_{i})^{n_{i}-1}}{(n_{i}-1)!}&\cdot\idotsint\limits_{\mathbf{I}}\mathbf{1}_{\\{\alpha_{i}\\}}(u_{i})\cdot\mathbf{1}_{b_{i}=L}\prod_{j\neq
i}\frac{(x_{j}-u_{j})^{n_{j}-1}}{(n_{j}-1)!}\hat{\chi}_{B,j}(u_{j})\
d\mu_{\mathbf{b}}(u_{1},\dots,u_{d}).\end{split}$

Since the last integral does not depend on $x_{i}$, we obtain that the above
formula represents a pseudo-polynomial of degree $(n_{1}-1,\dots,n_{d}-1)$.

It follows that the expression (19) is a sum of a pseudo-polynomial of degree
$(n_{1}-1,\dots,n_{d}-1)$ and the summand corresponding to $B=\emptyset$,
namely
$\idotsint\limits_{\mathbf{I}}\prod_{j=1}^{d}\frac{(x_{j}-u_{j})^{n_{j}-1}}{(n_{j}-1)!}\chi_{\alpha_{j},x_{j}}^{r}(u_{j})\
d\mu_{\mathbf{b}}(u_{1},\dots,u_{d})$. Consequently, the function $f$ is of
the form (22), where $\mu=\sum_{\mathbf{b}\in\\{L,r\\}^{d}}\mu_{\mathbf{b}}$.
The theorem is proved. ∎

Using the same method as in the proof of Theorem 33, we can prove the
following corollary.

######  Corollary 34.

Let $I_{1},I_{2},\dots,I_{d}$ be open intervals (bounded or unbounded) and
$f\colon\mathbf{I}\to\mathbb{R}$ be a function. Let
$\boldsymbol{\alpha}=(\alpha_{1},\alpha_{2},\dots,\alpha_{d})\in\mathbf{I}$
and $\mathbf{n}=(n_{1},n_{2},\dots,n_{d})\in\mathbb{N}^{d}$ be such that
$n_{i}\geq 1$ for $i=1,2,\dots,d$. We put
$\mathbf{B}=\\{(b_{1},\dots,b_{d})\in\\{L,r\\}^{d}:b_{i}=r$ if $n_{i}\geq
2,i=1,\dots,d\\}$.

Then $f$ is box-$\mathbf{n}$-convex, if and only if it is of the form

(24)
$f(x_{1},\dots,x_{d})=W(x_{1},\dots,x_{d})+\sum_{\mathbf{b}\in\mathbf{B}}\idotsint\limits_{\mathbf{I}}\prod_{j=1}^{d}\frac{(x_{j}-u_{j})^{n_{j}-1}}{(n_{j}-1)!}\chi_{\alpha_{j},x_{j}}^{b_{j}}(u_{j})\
d\mu_{\mathbf{b}}(u_{1},\dots,u_{d}),$

where $W\colon\mathbf{I}\to\mathbb{R}$ is a pseudo-polynomial of degree
$(n_{1}-1,\dots,n_{d}-1)$ and measures $\mu_{\mathbf{b}}$
$(\mathbf{b}\in\mathbf{B})$ satisfy conditions (ii) and (iii) of Theorem 30
$($here we use the standard convention that $0^{0}=1)$.

######  Remark 35.

The representation (24) is a counterpart of the representation (22) in the
case when some (not necessarily all) of $n_{i}$’s are greater than $1$. In
particular, Theorem 33 is a special case of Corollary 34. It can be shown that
for given $f$ and $\boldsymbol{\alpha}$ the pseudo-polynomial $W$ and the
measures $\mu_{\mathbf{b}}$ ($\mathbf{b}\in\mathbf{B}$) are uniquely
determined.

######  Remark 36.

In the above results, we focused on the box-$\mathbf{n}$-convex functions, in
the case when $n_{i}\geq 1$ for all $i=1,2,\dots,d$. We can also consider the
case when $n_{i}=0$ for some $i$.

Let $f\colon\mathbf{I}\to\mathbb{R}$ and
$\mathbf{n}=(n_{1},n_{2},\dots,n_{d})\in\mathbb{N}^{d}$ be such that
$n_{i}\geq 0$ for $i=1,2,\dots,d$. Let $A=\\{i=1,2,\dots,d:n_{i}\geq 1\\}$.
Then the function $f$ is box-$\mathbf{n}$-convex if and only if for every
$\mathbf{z}\in\mathbf{I}_{A^{\prime}}$ the function $f_{A}^{\mathbf{z}}$ is
box-$\mathbf{n}_{A}$-convex. Since all the coordinates of $\mathbf{n}_{A}$ are
positive, the box-$\mathbf{n}_{A}$-convexity is characterized in Corollary 34
and the preceding results.

Let $x_{+}=\max\\{0,x\\}$ and $x_{-}=\max\\{0,-x\\}$, then $x=x_{+}-x_{-}$
$(x\in{\mathbb{R}})$. It is not difficult to prove the following version of
Theorem 33, which is a $d$-dimensional counterpart of the well-known integral
spline representation of $n$-convex functions in one dimension.

######  Theorem 37.

Let $I_{1},I_{2},\dots,I_{d}$ be open intervals (bounded or unbounded). Let
$\mathbf{n}=(n_{1},n_{2},\dots,n_{d})\in\mathbb{N}^{d}$ be such that
$n_{i}\geq 2$ for $i=1,2,\dots,d$.

Let $f\colon\mathbf{I}\to\mathbb{R}$ be a function. Then $f$ is
box-$\mathbf{n}$-convex, if and only if for every
$\boldsymbol{\alpha}=(\alpha_{1},\alpha_{2},\dots,\alpha_{d})\in\mathbf{I}$,
there exist a pseudo-polynomial $W_{\boldsymbol{\alpha}}$ of degree
$(n_{1}-1,\dots,n_{d}-1)$ and a Borel measure $\mu$ on $\mathbf{I}$, which is
finite on convex sets, such that $f$ is of the form

(25) $f(x_{1},\dots,x_{d})=W_{\boldsymbol{\alpha}}(x_{1},\dots,x_{d})\\\
+\sum_{A\subset\\{1,2,\ldots,d\\}}\idotsint\limits_{\mathbf{I}_{A,\boldsymbol{\alpha}}}\prod_{j\in
A^{\prime}}\frac{(x_{j}-u_{j})_{+}^{n_{j}-1}}{(n_{j}-1)!}\prod_{j\in
A}(-1)^{n_{j}}\frac{(x_{j}-u_{j})_{-}^{n_{j}-1}}{(n_{j}-1)!}\
d\mu(u_{1},\dots,u_{d}),$

where
$\mathbf{I}_{A,\boldsymbol{\alpha}}=\\{(u_{1},\ldots,u_{d})\in\textbf{I}\colon
u_{j}\leq\alpha_{j}\>if\>j\in A\>and\>u_{j}>\alpha_{j}\>if\>j\in
A^{\prime}\\}$.

######  Theorem 38.

Let $I_{1},I_{2},\dots,I_{d}$ be open intervals (bounded or unbounded). Let
$\mathbf{n}=(n_{1},n_{2},\dots,n_{d})\in\mathbb{N}^{d}$ be such that
$n_{i}\geq 2$ for $i=1,2,\dots,d$.

Let $f\colon\mathbf{I}\to\mathbb{R}$ be a function. Then $f$ is
box-$\mathbf{n}$-convex, if and only if for all
$\boldsymbol{\alpha}=(\alpha_{1},\alpha_{2},\dots,\alpha_{d})\in\mathbf{I}$,
and
$\mathbf{J}^{\boldsymbol{\alpha}}=\mathbf{I}\cap\prod_{i=1}^{d}(\alpha_{i},\infty)$,
we have

$f(x_{1},\dots,x_{d})=W_{\boldsymbol{\alpha}}(x_{1},\dots,x_{d})+\idotsint\limits_{\mathbf{J}^{\boldsymbol{\alpha}}}\prod_{j=1}^{d}\frac{(x_{j}-u_{j})_{+}^{n_{j}-1}}{(n_{j}-1)!}\
d\mu_{\boldsymbol{\alpha}}(u_{1},\dots,u_{d}),$

for $(x_{1},\dots,x_{d})\in\mathbf{J}^{\boldsymbol{\alpha}}$, where
$W_{\boldsymbol{\alpha}}\colon\mathbf{J}^{\boldsymbol{\alpha}}\to\mathbb{R}$
is a pseudo-polynomial of degree $(n_{1}-1,\dots,n_{d}-1)$, and
$\mu_{\boldsymbol{\alpha}}$ is a Borel measure on $\mathbf{I}$, which is
finite on convex sets.

###### Proof.

The implication ($\Rightarrow$) is an immediate consequence of Theorem 33 (we
take $W_{\boldsymbol{\alpha}}=W|_{\mathbf{J}^{\boldsymbol{\alpha}}}$ and
$\mu_{\boldsymbol{\alpha}}=\mu$).

We show the implication ($\Leftarrow$). By Theorem 33, the function

$g(x_{1},x_{2},\dots,x_{d})=\idotsint\limits_{\mathbf{I}}\prod_{j=1}^{d}\frac{(x_{j}-u_{j})^{n_{j}-1}}{(n_{j}-1)!}\chi_{\alpha_{j},x_{j}}^{r}(u_{j})\
d\mu_{\boldsymbol{\alpha}}(u_{1},\dots,u_{d})$

is box-$\mathbf{n}$-convex on $\mathbf{I}$. As a consequence, it is also
box-$\mathbf{n}$-convex on $\mathbf{J}^{\boldsymbol{\alpha}}$. Consequently,
the function $f|_{\mathbf{J}^{\boldsymbol{\alpha}}}=W_{\boldsymbol{\alpha}}+g$
is box-$\mathbf{n}$-convex.

Since for every $\boldsymbol{\alpha}\in\mathbf{I}$, $f$ is
box-$\mathbf{n}$-convex on $\mathbf{J}^{\boldsymbol{\alpha}}$, we obtain that
$f$ is box-$\mathbf{n}$-convex on $\mathbf{I}$. ∎

##  9\. Box-$\mathbf{n}$-convex orders.

By the analogy to the $n$-convex orders and the box-$(m,n)$-convex orders [2,
12, 19], we will define the box-$\mathbf{n}$-convex orders.

First, let us recall the definition of the $n$-convex order.

######  Definition 7.

Let $X$ and $Y$ be two random variables that take on values in the interval
$I\subset{\mathbb{R}}$. Then $X$ is said to be smaller than $Y$ in the
$n$-convex (respectively, $n$-concave) order, denoted by $X\leq_{n\mathchar
45\relax cx}Y$ ($X\leq_{n\mathchar 45\relax cv}Y$), if

$\mathbb{E}\,f(X)\leq\mathbb{E}\,f(Y)$

for all $n$-convex ($n$-concave) functions $f:I\to\mathbb{R}$, for which the
expectations exist.

Many properties of the $n$-convex order can be found in [2].

######  Theorem 39.

Let $X$ and $Y$ be two $I$-valued random variables such that
$\mathbb{E}\,|X|^{n}<\infty$ and $\mathbb{E}\,|Y|^{n}<\infty$. Then
$X\leq_{n\mathchar 45\relax cx}Y$ if, and only if,

  * a)

$\mathbb{E}\,X^{k}=\mathbb{E}\,Y^{k}$, $k=1,\ldots,n$,

  * b)

$\mathbb{E}\,(X-t)^{n}_{+}\leq\mathbb{E}\,(Y-t)^{n}_{+}$ for all $t\in I$.

######  Theorem 40.

Let $X$ and $Y$ be two $I$-valued random variables such that
$\mathbb{E}\,|X|^{n}<\infty$ and $\mathbb{E}\,|Y|^{n}<\infty$. Then
$X\leq_{n\mathchar 45\relax cx}Y$ if, and only if,

  * a)

$\mathbb{E}\,X^{k}=\mathbb{E}\,Y^{k}$, $k=1,\ldots,n$,

  * b’)

$\mathbb{E}\,(-1)^{n+1}(X-t)^{n}_{-}\leq\mathbb{E}\,(-1)^{n+1}(Y-t)^{n}_{-}$
for all $t\in I$.

It is not difficult to prove the following lemma.

######  Lemma 41.

Let $x,u\in{\mathbb{R}}$, $n\in{\mathbb{N}}$, then
$(x-u)^{n}_{+}=(x-u)^{n}+(-1)^{n+1}(x-u)^{n}_{-}$.

Note, that Theorem 40 easily follows from Theorem 39 and Lemma 41.

By analogy to the $n$-convex order for random variables, one can define the
$n$-convex order for the signed measures $\gamma_{1}$, $\gamma_{2}$ on $I$,
which have a finite variation: $\gamma_{1}\leq_{n\mathchar 45\relax
cx}\gamma_{2}$ if $\int_{I}f(x)d\gamma_{1}(x)\leq\int_{I}f(x)d\gamma_{2}(x)$
for all for all $n$-convex functions $f\colon I\to\mathbb{R}$, provided the
integrals exist. Similarly one can define the $n$-concave order
$(\leq_{n\mathchar 45\relax cv})$ for signed measures.

By Theorem 39 (cf [17]), it follows the following characterization of
$n$-convex orders for the signed measures.

######  Theorem 42.

Let $\gamma$ be a signed measure on $I$, which has a finite variation and such
that $\int_{I}|x|^{n}d|\gamma|(x)<\infty$. Then $\gamma\geq_{n\mathchar
45\relax cx}0$ if, and only if,

  * a)

$\int_{I}x^{k}d\gamma(x)=0$, $k=1,\ldots,n$,

  * b)

$\int_{I}(x-u)^{n}_{+}d\gamma(x)\geq 0$ for all $u\in I$.

######  Remark 43.

By Theorem 40, it follows, that condition (b) in Theorem 42, can be replaced
by

  * b’)

$\int_{I}(-1)^{n+1}(x-u)^{n}_{-}d\gamma(x)\geq 0$ for all $u\in I$.

######  Remark 44.

Similarly, one can give characterizations of $n$-concave orders if conditions
(b) and (b’) (in Theorems 39, 40, 42 and Remark 43) hold with the sign
reversed.

By analogy to the $n$-convex orders and the box-$(m,n)$-convex orders, we
define the box-$\mathbf{n}$-convex orders.

######  Definition 8.

Let $(X_{1},\ldots,X_{d})$ and $(Y_{1},\ldots,Y_{d})$ be two
$\mathbf{I}$-valued random vectors. Then $(X_{1},\ldots,X_{d})$ is said to be
smaller then $(Y_{1},\ldots,Y_{d})$ in the box-$\mathbf{n}$-convex order,
denoted by $(X_{1},\ldots,X_{d})\prec_{box\mathchar
45\relax\mathbf{n}\mathchar 45\relax cx}(Y_{1},\ldots,Y_{d})$, if

(26) $\mathbb{E}\,f(X_{1},\ldots,X_{d})\leq\mathbb{E}\,f(Y_{1},\ldots,Y_{d})$

for all continuous box-$\mathbf{n}$-convex functions
$f\colon\mathbf{I}\to\mathbb{R}$, provided the expectations exist.

######  Theorem 45.

Let $(X_{1},\ldots,X_{d})$ and $(Y_{1},\ldots,Y_{d})$ be two
$\mathbf{I}$-valued random vectors such that
$\mathbb{E}\,|X_{1}|^{n_{1}-1}\ldots|X_{d}|^{n_{d}-1}<\infty$ and
$\mathbb{E}\,|Y_{1}|^{n_{1}-1}\ldots|Y_{d}|^{n_{d}-1}<\infty$. Then

(27) $(X_{1},\ldots,X_{d})\prec_{box\mathchar 45\relax\mathbf{n}\mathchar
45\relax cx}(Y_{1},\ldots,Y_{d})$

if, and only if, the following conditions are satisfied

  * a)

(28) $\mathbb{E}\,W(X_{1},\ldots,X_{d})=\mathbb{E}\,W(Y_{1},\ldots,Y_{d})$

for all continuous pseudo-polynomials $W$ of degree $(n_{1}-1,\ldots
n_{d}-1)$, provided the expectations exist,

  * b)

(29) $\mathbb{E}\,\prod_{j\in
A^{\prime}}\frac{(X_{j}-u_{j})_{+}^{n_{j}-1}}{(n_{j}-1)!}\prod_{j\in
A}(-1)^{n_{j}}\frac{(X_{j}-u_{j})_{-}^{n_{j}-1}}{(n_{j}-1)!}\leq\mathbb{E}\,\prod_{j\in
A^{\prime}}\frac{(Y_{j}-u_{j})_{+}^{n_{j}-1}}{(n_{j}-1)!}\prod_{j\in
A}(-1)^{n_{j}}\frac{(Y_{j}-u_{j})_{-}^{n_{j}-1}}{(n_{j}-1)!}$

for all $A\subset\\{1,2,\ldots,d\\}$ and $(u_{1},\ldots,u_{d})\in\mathbf{I}$.

###### Proof.

We show the implication ($\Rightarrow$). Let $W$ be a continuous pseudo-
polynomial of degree $(n_{1}-1,\ldots n_{d}-1)$. By Lemma 11, $W$ is
box-$\mathbf{n}$-affine. Then taking in (26) $f=W$ and next $f=-W$, we obtain
(28).

Let $A\subset\\{1,2,\ldots,d\\}$ and
$\mathbf{u}=(u_{1},\ldots,u_{d})\in\mathbf{I}$ be fixed. Then there exists
$(\alpha_{1},\ldots,\alpha_{d})$ $\in\mathbf{I}$ such that $\alpha_{j}<u_{j}$
if $j\in A^{\prime}$ and $\alpha_{j}>u_{j}$ if $j\in A$, consequently
$\mathbf{u}\in\mathbf{I}_{A,(\alpha_{1},\ldots,\alpha_{d})}$. Let
$f_{A,\mathbf{u}}\colon\mathbf{I}\to\mathbb{R}$ be the function given by the
formula

$f_{A,\mathbf{u}}(x_{1},\ldots,x_{d})=\prod_{j\in
A^{\prime}}\frac{(x_{j}-u_{j})_{+}^{n_{j}-1}}{(n_{j}-1)!}\prod_{j\in
A}(-1)^{n_{j}}\frac{(x_{j}-u_{j})_{-}^{n_{j}-1}}{(n_{j}-1)!}$

Then, by Theorem 37, the function $f_{A,\mathbf{u}}$ is
box-$\mathbf{n}$-convex and it is of the form (25) with
$W_{\boldsymbol{\alpha}}=0$ and $\mu=\delta_{(u_{1},\ldots,u_{d})}$. Then by
(26), taking $f=f_{A,\mathbf{u}}$, we obtain (29).

We pass to the proof of the implication ($\Leftarrow$). Let
$f\colon\mathbf{I}\to\mathbb{R}$ be a continuous box-$\mathbf{n}$-convex
function. Let
$\boldsymbol{\alpha}=(\alpha_{1},\alpha_{2},\dots,\alpha_{d})\in\mathbf{I}$.
Then, by Theorem 37, the function $f$ is of the form (25). Then

(30)
$\mathbb{E}\,f(X_{1},\dots,X_{d})=\mathbb{E}\,W_{\boldsymbol{\alpha}}(X_{1},\dots,X_{d})\\\
+\sum_{A\subset\\{1,2,\ldots,d\\}}\idotsint\limits_{\mathbf{I_{A,\alpha}}}\mathbb{E}\,\prod_{j\in
A^{\prime}}\frac{(X_{j}-u_{j})_{+}^{n_{j}-1}}{(n_{j}-1)!}\prod_{j\in
A}(-1)^{n_{j}}\frac{(X_{j}-u_{j})_{-}^{n_{j}-1}}{(n_{j}-1)!}\
d\mu(u_{1},\dots,u_{d}).$

Inequality (26) now follows from (28), (29), using identity (30), and a
similar identity involving $(Y_{1},\dots,Y_{d})$. Thus inequality (27) is
satisfied.

∎

By analogy to box-$\mathbf{n}$-convex orders for random vectors, we define the
box-$\mathbf{n}$-convex order for signed measures $\gamma_{1}$, $\gamma_{2}$
on $\mathbf{I}$, which have a finite variation: $\gamma_{1}\prec_{box\mathchar
45\relax\mathbf{n}\mathchar 45\relax cx}\gamma_{2}$ if

$\idotsint\limits_{\mathbf{I}}f(x_{1},\dots,x_{d})d\gamma_{1}(x_{1},\dots,x_{d})\leq\idotsint\limits_{\mathbf{I}}f(x_{1},\dots,x_{d})d\gamma_{2}(x_{1},\dots,x_{d})$

for all continuous box-$\mathbf{n}$-convex functions
$f\colon\mathbf{I}\to\mathbb{R}$, provided the integrals exist. From Theorem
45, it follows the following characterization of box-$\mathbf{n}$-convex order
for signed measures.

######  Theorem 46.

Let $\gamma$ be a signed measure on $\mathbf{I}$ which has a finite variation
and such that
$\idotsint\limits_{\mathbf{I}}|x_{1}|^{n_{1}-1},\dots,|x_{d}|^{n_{d}-1}d|\gamma|(x_{1},\dots,x_{d})<\infty$.
Then

$\idotsint\limits_{\mathbf{I}}f(x_{1},\dots,x_{d})\>d\gamma(x_{1},\dots,x_{d})\geq
0$

for all continuous box-$\mathbf{n}$-convex functions
$f\colon\mathbf{I}\to\mathbb{R}$ if, and only if,

  * a)

(31)
$\idotsint\limits_{\mathbf{I}}W(x_{1},\dots,x_{d})\>d\gamma(x_{1},\dots,x_{d})=0$

for all continuous pseudo-polynomials $W$ of order $(n_{1}-1,\ldots n_{d}-1)$,
provided the integral exists,

  * b)

(32) $\idotsint\limits_{\mathbf{I}}\prod_{j\in
A^{\prime}}\frac{(x_{j}-u_{j})_{+}^{n_{j}-1}}{(n_{j}-1)!}\prod_{j\in
A}(-1)^{n_{j}}\frac{(x_{j}-u_{j})_{-}^{n_{j}-1}}{(n_{j}-1)!}\>d\gamma(x_{1},\dots,x_{d})\geq
0$

for all $A\subset\\{1,2,\ldots,d\\}$ and $(u_{1},\ldots,u_{d})\in\mathbf{I}$.

By Theorem 46, we obtain the following characterization of
box-$\mathbf{n}$-convex order for the signed measure
$\gamma=\gamma_{1}\otimes,\ldots,\otimes\gamma_{d}$, which is a product
measure of $\gamma_{1},\ldots,\gamma_{d}$.

######  Theorem 47.

Let $\gamma_{i}$ be non zero signed finite Borel measures on $I_{i}$, such
that $\int_{I_{i}}|x_{i}|^{n_{i}-1}\>d|\gamma_{i}|(x_{i})<\infty$,
$i=1,\ldots,d$. Then

(33)
$\idotsint\limits_{\mathbf{I}}f(x_{1},\dots,x_{d})\>d\gamma_{1}(x_{1})\ldots\>d\gamma_{d}(x_{d})\geq
0$

for all continuous box-$\mathbf{n}$-convex functions
$f\colon\mathbf{I}\to\mathbb{R}$ (provided the integral exists) if, and only
if ,

  * a)

for all $j=1,\ldots,d$ either $\gamma_{j}\geq_{(n_{j}-1)\mathchar 45\relax
cx}0$ or $\gamma_{j}\geq_{(n_{j}-1)\mathchar 45\relax cv}0$,

  * b)

the number of those $j$ for which $\gamma_{j}\geq_{(n_{j}-1)\mathchar 45\relax
cv}0$ is even.

###### Proof.

We show the implication ($\Rightarrow$). By Theorem 46 (a)

$\idotsint\limits_{\mathbf{I}}W(x_{1},\dots,x_{d})\>\>d\tau_{1}(x_{1})\ldots\>d\gamma_{d}(x_{d})=0$

for all continuous pseudo-polynomials of degree $(n_{1}-1,\ldots,n_{d}-1)$ of
the form

(34)
$W(x_{1},x_{2},\dots,x_{d})=\sum_{i=1}^{d}\sum_{k=0}^{n_{i}-1}A_{ik}(\mathbf{x}_{\\{i\\}^{\prime}})x_{i}^{k},$

provided the integral exists. Then, for all $i=1,\ldots,d$ and
$k=0,\ldots,n_{i}-1$

(35)
$\int_{I_{i}}x_{i}^{k}\>d\gamma_{i}(x_{i})\idotsint\limits_{I_{1}\times\ldots\times
I_{i-1}\times I_{i-1}\times\ldots\times
I_{d}}A_{ik}(\mathbf{x}_{\\{i\\}^{\prime}})\>d\gamma_{1}(x_{1})\ldots\>d\gamma_{i-1}(x_{i-1})\
d\gamma_{i+1}(x_{1})\ldots\>d\gamma_{d}(x_{d})=0.$

Since $\gamma_{i}$, $i=1,\ldots,d$ are non-zero signed measures, it follows
that there exists the functions $A_{ik}$, such that the second integral in the
above expression is non-zero. Then by (35), we obtain

(36) $\int_{I_{i}}x_{i}^{k}\>d\gamma_{i}(x_{i})=0,\quad
i=1,\ldots,d,\>k=0,1,\ldots,n_{i}-1.$

By (32), for $A=\emptyset$, we have

(37)
$\prod_{j=1}^{d}\int_{I_{j}}\frac{(x_{j}-u_{j})_{+}^{n_{j}-1}}{(n_{j}-1)!}\>d\gamma_{j}(x_{j})\geq
0$

for all $(u_{1},\ldots,u_{d})\in\mathbf{I}$. We denote

$H(\gamma_{j},u_{j})=\int_{I_{j}}\frac{(x_{j}-u_{j})_{+}^{n_{j}-1}}{(n_{j}-1)!}\>d\gamma_{j}(x_{j}).$

Since $\gamma_{i}$, $i=1,\ldots,d$, are non-zero signed measures, it follows
that there exist $(v_{1},\ldots,v_{d})\in\mathbf{I}$ such that
$H(\gamma_{i},v_{i})\neq 0$ for $i=1,\dots,d$.

Let $i=1,\ldots,d$. By (37), $H(\gamma_{i},u_{i})\prod_{j\neq
i}H(\gamma_{j},v_{j})\geq 0$ for each $u_{i}\in I_{i}$. Consequently, either
$H(\gamma_{i},u_{i})\geq 0$ for all $u_{i}\in I_{i}$ or
$H(\gamma_{i},u_{i})\leq 0$ for all $u_{i}\in I_{i}$.

Moreover, by (37), we conclude, that the number of $j$’s such that
$H(\gamma_{j},u_{j})\leq 0$ for all $u_{j}\in I_{j}$, is even. Taking into
account (36), by Theorem 42 and Remark 44, ($\Rightarrow$) is proved.

We pass to the proof of the implication ($\Leftarrow$). Assume, that
conditions a) and b) are satisfied. By Theorem 42, we have that (36) is
satisfied, then taking into account (35), (34), we obtain (31) with
$\gamma=\gamma_{1}\otimes,\ldots,\otimes\gamma_{d}$.

By Theorem 42 and Remark 44 we have that for all $j=1,\ldots,d$, either
$H(\gamma_{j},u_{j})\geq 0$ for all $u_{j}\in I_{j}$ or
$H(\gamma_{j},u_{j})\leq 0$ for all $u_{j}\in I_{j}$, and the number of those
$j$ for which $H(\gamma_{j},u_{j})\leq 0$ for all $u_{j}\in I_{j}$, is even.

By Remark 43, we conclude that the function $H(\gamma_{j},u_{j})$ has the same
sign as the function

$S(\gamma_{j},u_{j})=\int_{I_{j}}(-1)^{n_{j}-1}\frac{(x_{j}-u_{j})_{+}^{n_{j}-1}}{(n_{j}-1)!}\>d\gamma_{j}(x_{j}).$

This implies that for each $A\subset\\{1,\dots,d\\}$ and
$(u_{1},\dots,u_{d})\in\mathbf{I}$, we have

$\prod_{j\in A^{\prime}}H(\gamma_{j},u_{j})\prod_{j\in
A}S(\gamma_{j},u_{j})\geq 0.$

Consequently, (32) is satisfied for
$\gamma=\gamma_{1}\otimes,\ldots,\otimes\gamma_{d}$. By Theorem 46, this
completes the proof of ($\Leftarrow$). ∎

By Theorem 42 and Remark 43, we obtain that Theorem 47 can be rewritten in the
following form.

######  Theorem 48.

Let $\gamma_{i}$ be non zero signed finite Borel measures on $I_{i}$, such
that $\int_{I_{i}}|x_{i}|^{n_{i}-1}\>d|\gamma_{i}|(x_{i})<\infty$,
$i=1,\ldots,d$. Then

$\idotsint\limits_{\mathbf{I}}f(x_{1},\dots,x_{d})\>d\gamma_{1}(x_{1})\ldots\>d\gamma_{d}(x_{d})\geq
0$

for all continuous box-$\mathbf{n}$-convex functions
$f\colon\mathbf{I}\to\mathbb{R}$ (provided the integral exists) if, and only
if ,

  * a)

(38) $\int_{I_{i}}x_{i}^{k}\>d\gamma_{i}(x_{i})=0,\quad
i=1,\ldots,d,\>k=0,1,\ldots,n_{i}-1.$

  * b)

$\prod_{j=1}^{d}\int_{I_{j}}\frac{(x_{j}-u_{j})_{+}^{n_{j}-1}}{(n_{j}-1)!}\>d\gamma_{j}(x_{j})\geq
0$

for all $(u_{1},\ldots,u_{d})\in\mathbf{I}$.

From Theorem 47, it follows immediately the following theorem.

######  Theorem 49.

Let $\mu_{i}$, $\nu_{i}$ be probability measures on $I_{i}$ such that
$\int_{I_{i}}|x_{i}|^{n_{i}-1}$ $d\mu_{i}(x_{i})<\infty$,
$\int_{I_{i}}|x_{i}|^{n_{i}-1}\>d\nu_{i}(x_{i})<\infty$ and
$\nu_{i}\leq_{(n_{j}-1)\mathchar 45\relax cx}\mu_{i}$, $i=1,\ldots,d$. Then

$\idotsint\limits_{\mathbf{I}}f(x_{1},\dots,x_{d})\>d(\mu_{1}-\nu_{1})(x_{1})\ldots\>d(\mu_{d}-\nu_{d})(x_{d})\geq
0$

for all continuous box-$\mathbf{n}$-convex functions
$f\colon\mathbf{I}\to\mathbb{R}$ (provided the integral exists).

##  10\. The Hermite-Hadamard, Jensen and Raşa inequalities

We recall the classical Hermite-Hadamard and Jensen inequalities.

######  Proposition 7.

Let $f\colon I\to\mathbb{R}$ be a convex function defined on a real interval
$I$ and $a,b\in I$ with $a<b$. The following double inequality

$f\left(\frac{a+b}{2}\right)\leq\frac{1}{b-a}\cdot\int_{a}^{b}f(x)\,dx\leq\frac{f(a)+f(b)}{2}$

is known as the Hermite-Hadamard inequality for convex functions (see [3]),
which is equivalent to the convex ordering relations (see [16])

$\delta_{(a+b)/2}\leq_{1\mathchar 45\relax
cx}\>\frac{1}{b-a}\chi_{[a,b]}(x)\,dx\leq_{1\mathchar 45\relax
cx}\>\frac{1}{2}(\delta_{a}+\delta_{b}).$

######  Proposition 8.

One of the most familiar and elementary inequalities in the probability theory
is the Jensen inequality:

(39) $f\bigl{(}\mathbb{E}\,X\bigr{)}\leq\mathbb{E}\,f(X),$

where the function $f$ is convex over the convex hull of the range of the
random variable $X$ (see [1]). Inequality (39) is equivalent to the convex
ordering relation (see [16])

$\delta_{\mathbb{E}\,X}\leq_{1\mathchar 45\relax cx}\>\mu_{X}.$

In this paper, we give some Hermite-Hadamard, Jensen and Raşa inequalities for
box-$\mathbf{n}$-convex functions.

Note that Theorem 49 is equivalent to the following theorem.

######  Theorem 50.

Let $X_{i}$, $Y_{i}$ be $I_{i}$ valued random variables such that
$\mathbb{E}\,|X_{i}|^{n_{i-1}}<\infty$, $\mathbb{E}\,|Y_{i}|^{n_{i-1}}<\infty$
and $X_{i}\leq_{(n_{j}-1)\mathchar 45\relax cx}Y_{i}$, $i=1,\ldots,d$. For
each $A\subset\\{1,\ldots,d\\}$, by $Z_{1,A},\ldots,Z_{d,A}$ we denote
independent random variables such that

(40) $\mu_{Z_{i,A}}=\left\\{\begin{aligned} \mu_{Y_{i}}\quad&\ \text{if
}i\notin A,\\\ \mu_{X_{i}}\quad&\ \text{if }i\in A.\end{aligned}\right.$

Then

(41)
$\sum_{A\subset\\{1,\ldots,d\\}}(-1)^{|A|}\mathbb{E}\,f(Z_{1,A},\ldots,Z_{d,A})\geq
0,$

for all continuous box-$\mathbf{n}$-convex functions
$f\colon\mathbf{I}\to\mathbb{R}$ (provided the expectations exist).

From Proposition 7 and Theorem 50, we obtain the following Hermite-Hadamard
type inequalities for box-$(2,\ldots,2)$-convex functions.

######  Theorem 51.

Let $a_{i},b_{i}\in I_{i}$ with $a_{i}<b_{i}$, $i=1,\ldots,d$. Then, for all
continuous box-$(2,\ldots,2)$-convex functions
$f\colon\mathbf{I}\to\mathbb{R}$,

  * (a)

the first Hermite-Hadamard inequality is given by (41), with

$\mu_{X_{i}}=\delta_{(a_{i}+b_{i})/2}\quad\mu_{Y_{i}}=\frac{1}{b_{i}-a_{i}}\chi_{[a_{i},b_{i}]}(x)\,dx,\quad(i=1,\ldots,d),$

  * (b)

the second Hermite-Hadamard inequalityis given by (41), with

$\mu_{X_{i}}=\frac{1}{b_{i}-a_{i}}\chi_{[a_{i},b_{i}]}(x)\,dx,\quad\mu_{Y_{i}}=\frac{1}{2}(\delta_{a_{i}}+\delta_{b_{i}})\quad(i=1,\ldots,d).$

For $d=3$, we obtain the following first Hermite-Hadamard inequality.

######  Theorem 52.

Let $a_{i},b_{i}\in I_{i}$ with $a_{i}<b_{i}$, $i=1,2,3$. Then, for all
continuous box-$(2,2,2)$-convex functions $f\colon\mathbf{I}\to\mathbb{R}$, we
have

the first Hermite-Hadamard inequality

$\frac{1}{(b_{1}-a_{1})(b_{2}-a_{2})(b_{3}-a_{3})}\int_{a_{1}}^{b_{1}}\int_{a_{2}}^{b_{2}}\int_{a_{3}}^{b_{3}}f(x,y,z)dz\,dy\,dx-\frac{1}{(b_{2}-a_{2})(b_{3}-a_{3})}\int_{a_{2}}^{b_{2}}\int_{a_{3}}^{b_{3}}f\left(\frac{a_{1}+b_{1}}{2},y,z\right)dz\,dy\\\
\\\
-\frac{1}{(b_{1}-a_{1})(b_{3}-a_{3})}\int_{a_{1}}^{b_{1}}\int_{a_{3}}^{b_{3}}f\left(x,\frac{a_{2}+b_{2}}{2},z\right)dz\,dx-\frac{1}{(b_{1}-a_{1})(b_{2}-a_{2})}\int_{a_{1}}^{b_{1}}\int_{a_{2}}^{b_{2}}f\left(x,y,\frac{a_{3}+b_{3}}{2}\right)dy\,dx\\\
\\\
+\frac{1}{(b_{1}-a_{1})}\int_{a_{1}}^{b_{1}}f\left(x,\frac{a_{2}+b_{2}}{2},\frac{a_{3}+b_{3}}{2}\right)dx+\frac{1}{(b_{2}-a_{2})}\int_{a_{2}}^{b_{2}}f\left(\frac{a_{1}+b_{1}}{2},y,\frac{a_{3}+b_{3}}{2}\right)dy\\\
\\\
+\frac{1}{(b_{3}-a_{3})}\int_{a_{3}}^{b_{3}}f\left(\frac{a_{1}+b_{1}}{2},\frac{a_{2}+b_{2}}{2},z\right)dz-f\left(\frac{a_{1}+b_{1}}{2},\frac{a_{2}+b_{2}}{2},\frac{a_{3}+b_{3}}{2}\right)\geq
0.$

From Proposition 7 and Theorem 50, we obtain the following Jensen type
inequalities for box-$\mathbf{n}$-convex functions.

######  Theorem 53.

Let $X_{i}$ be $I_{i}$ valued random variables, $i=1,\ldots,d$. Then, for all
continuous box-$(2,\ldots,2)$-convex functions
$f\colon\mathbf{I}\to\mathbb{R}$, we have the Jensen inequality

$\sum_{A\subset\\{1,\ldots,d\\}}(-1)^{|A|}\mathbb{E}\,f(Z_{1,A},\ldots,Z_{d,A})\geq
0,$

where $Z_{1,A},\ldots,Z_{d,A}$ are independent random variables such that

$\mu_{Z_{i,A}}=\left\\{\begin{aligned} \mu_{X_{i}}\quad&\ \text{if }i\notin
A,\\\ \delta_{\>\mathbb{E}\,X_{i}}\quad&\ \text{if }i\in
A.\end{aligned}\right.$

For $d=3$, we obtain the following Jensen inequality.

######  Theorem 54.

Let $X,Y,Z$ be $I_{1},I_{2},I_{3}$ valued random variables, respectively.
Then, for all continuous box-$(2,2,2)$-convex functions
$f\colon\mathbf{I}\to\mathbb{R}$, we have the Jensen inequality

$\mathbb{E}\,f(X,Y,Z)-\mathbb{E}\,f(\mathbb{E}\,X,Y,Z)-\mathbb{E}\,f(X,\mathbb{E}\,Y,Z)-\mathbb{E}\,f(X,Y,\mathbb{E}\,Z)\\\
+\mathbb{E}\,f(X,\mathbb{E}\,Y,\mathbb{E}\,Z)+\mathbb{E}\,f(\mathbb{E}\,X,Y,\mathbb{E}\,Z)+\mathbb{E}\,f(\mathbb{E}\,X,\mathbb{E}\,Y,Z)-f(\mathbb{E}\,X,\mathbb{E}\,Y,\mathbb{E}\,Z)\geq
0.$

If the random variables have discrete distributions, then the above Jensen
inequality has the following form.

######  Theorem 55.

Let $I_{1},I_{2},I_{3}$ be real intervals. Let $M,N,L>1$.
$x_{1},\ldots,x_{M}\in I_{1}$, $y_{1},\ldots,y_{N}\in I_{2}$,
$z_{1},\ldots,z_{L}\in I_{3}$,
$\alpha_{1},\ldots,\alpha_{M},\beta_{1},\ldots,\beta_{N},\gamma_{1},\ldots,\gamma_{L}\geq
0$,
$\alpha_{1}+\ldots+\alpha_{M}=\beta_{1}+\ldots+\beta_{N}=\gamma_{1}+\ldots+\gamma_{L}=1$,
$\bar{x}=\alpha_{1}x_{1}+\ldots+\alpha_{M}x_{M}$,
$\bar{y}=\beta_{1}y_{1}+\ldots+\beta_{N}y_{N}$,
$\bar{z}=\gamma_{1}z_{1}+\ldots+\gamma_{L}z_{L}$. Then for all continuous
box-$(2,2,2)$-convex functions

$\sum_{i=1}^{M}\sum_{j=1}^{N}\sum_{k=1}^{L}\alpha_{i}\beta_{j}\gamma_{k}f(x_{i},y_{j},z_{k})-\sum_{j=1}^{N}\sum_{k=1}^{L}\beta_{j}\gamma_{k}f(\bar{x},y_{j},z_{k})-\sum_{i=1}^{M}\sum_{k=1}^{L}\alpha_{i}\gamma_{k}f(x_{i},\bar{y},z_{k})-\sum_{i=1}^{M}\sum_{j=1}^{N}\alpha_{i}\beta_{j}f(x_{i},y_{j},\bar{z})\\\
+\sum_{i=1}^{M}\alpha_{i}f(x_{i},\bar{y},\bar{z})+\sum_{j=1}^{N}\beta_{j}f(\bar{x},y_{j},\bar{z})+\sum_{k=1}^{L}\gamma_{k}f(\bar{x},\bar{y},z_{k})-f(\bar{x},\bar{y},\bar{z})\geq
0.$

Now we recall the Raşa inequality. Its probabilistic version has the following
form (see [18], [13])

$(\nu-\mu)^{*2}\geq_{1\mathchar 45\relax cx}0,$

where $\mu=B(n,x)$ and $\nu=B(n,y)$ are the binomial distributions with
parameters $n\in\mathbb{N}$ and $x,y\in[0,1]$. By $(\nu-\mu)^{*2}$ we mean
$(\nu-\mu)*(\nu-\mu)$, where $*$ is the convolution of signed measures. In [9,
10, 11], we obtained some useful necessary and sufficient conditions for Borel
measures $\mu$ and $\nu$ to satisfy the following generalized Raşa inequality,
involving $q$-th convolutional power of the signed measure $\nu-\mu$:

$(\nu-\mu)^{*q}\geq_{(q-1)\mathchar 45\relax cx}0,\quad q\geq 2.$

In [12], we investigated the generalized Raşa inequality for
box-$(m,n)$-convex functions. In this paper, we study the generalized Raşa
inequality for box-$\mathbf{n}$-convex functions. We will need two lemmas.

######  Lemma 56 ([11], Lemma 5, p.5).

Let $\tau_{1},\ldots,\tau_{n}$ be signed measures on ${\mathbb{R}}$ with
finite variation, such that $\tau_{i}({\mathbb{R}})=0$ and
$\int_{-\infty}^{\infty}|x|^{n-1}|\tau_{i}|(dx)<\infty$, $i=1,\ldots,n$, Then

  * (a)

$\tau_{1}*\ldots*\tau_{n}({\mathbb{R}})=0$,

  * (b)

$\int_{-\infty}^{\infty}x^{k}\tau_{1}*\ldots*\tau_{n}(dx)=0$ for all integers
$0<k<n$.

######  Lemma 57 ([11], p.7).

Let $\tau_{1},\ldots,\tau_{q}$ be signed measures on ${\mathbb{R}}$ with
finite variation, such that $\tau_{i}({\mathbb{R}})$ $=0$, $i=1,\ldots,q$.
Then for all $A\in{\mathbb{R}}$

$\int_{-\infty}^{\infty}\frac{\bigl{(}x-A\bigr{)}^{q-1}_{+}}{(q-1)!}\>\tau_{1}*\ldots*\tau_{q}(dx)=\overline{F}_{\tau_{1}}*\overline{F}_{\tau_{2}}*\ldots*\overline{F}_{\tau_{q}}(A),$

where $\overline{F}_{\tau_{i}}(x)=\tau_{i}([x,\infty))$, $i=1,\ldots,q$,
$x\in{\mathbb{R}}$. On the right side of the above equality, the symbol $*$
denotes the convolution of functions on $\mathbb{R}$.

######  Theorem 58.

Let $\mathbf{n}=n_{1},\ldots,n_{d}$, $n_{i}\geq 2$, $i=1,\ldots,d$. Let
$\mu_{i},\nu_{i},$ be probability measures on ${\mathbb{R}}$, such that
$\int_{-\infty}^{\infty}|x|^{n_{i}-1}\mu_{i}(dx)<\infty$,
$\int_{-\infty}^{\infty}|x|^{n_{i}-1}\nu_{i}(dx)<\infty$ for $i=1,\dots,d$,
Then the following conditions are equivalent:

  * a)

the Raşa inequality for continuous box-$\mathbf{n}$-convex functions

$(\nu_{1}-\mu_{1})^{*n_{1}}\otimes\ldots\otimes(\nu_{d}-\mu_{d})^{*n_{d}}\succ_{box\mathchar
45\relax\mathbf{n}\mathchar 45\relax cx}0,$

  * b)

$(\overline{F}_{\nu_{1}}-\overline{F}_{\mu_{1}})^{*n_{1}}(A_{1})\times\ldots\times(\overline{F}_{\nu_{m}}-\overline{F}_{\mu_{m}})^{*n_{d}}(A_{d})\geq
0$

for all $A_{i}\in{\mathbb{R}}$, $i=1,\dots,d$.

###### Proof.

Let $\tau_{i}=\nu_{i}-\mu_{i}$, $i=1,\ldots,d$. Then $\tau_{i}$ are signed
measure such that $\tau_{i}({\mathbb{R}})=0$, $i=1,\ldots,d$, thus by Lemma 56
, it follows that for $\gamma_{i}=\tau_{i}^{*n_{i}}$

$\int_{-\infty}^{\infty}x_{i}^{k}\gamma_{i}(dx_{i})=0,\quad
k=0,1,\ldots,n_{i}-1,\quad i=1,\ldots,d.$

Then, by Theorem 48, we conclude that condition a) is equivalent to

$\prod_{j=1}^{d}\int_{I_{j}}\frac{(x_{j}-A_{j})_{+}^{n_{j}-1}}{(n_{j}-1)!}\>d\gamma_{j}(x_{j})\geq
0$

for all $(A_{1},\ldots,A_{d})\in\mathbf{I}$. Taking into account, that by
Lemma 57,

$\int_{I_{j}}\frac{(x_{j}-A_{j})_{+}^{n_{j}-1}}{(n_{j}-1)!}\>d\gamma_{j}(x_{j})=(\overline{F}_{\tau_{j}})^{*n_{j}}(A_{j}),\quad
j=1,\ldots,d,$

the theorem is proved. ∎

Similarly to [12], we define strongly box-$\mathbf{n}$-convex functions.

######  Definition 9.

We say that a function $f\colon\mathbf{I}\to\mathbb{R}$ is strongly
box-$\mathbf{n}$-convex functions with modulus $C\geq 0$, if the function
$g\colon\mathbf{I}\to\mathbb{R}$ given by the formula
$g(x_{1},\ldots,x_{d})=f(x_{1},\ldots,x_{d})-Cx_{1}^{n_{1}}\ldots
x_{d}^{n_{d}}$ $((x_{1},\ldots,x_{d})\in\mathbf{I})$ is
box-$\mathbf{n}$-convex.

Then, using the methods presented above and in [12], one can prove other the
Raşa, Jensen and Hermite-Hadamard inequalities for box-$\mathbf{n}$-convex
functions as well as for strongly box-$\mathbf{n}$-convex functions.

## References

  * [1] Billingsley P.: Probability and Measure, John Wiley & Sons, New York, (1995).
  * [2] Denuit M., Lefevre C., Shaked M.: The s-convex orders among real random variables, with applications. Math. Inequal. Appl.,1(1), 585–613, (1998).
  * [3] Dragomir S. S., Pearce C. E. M.: Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, (2000). (Online: http://rgmia.vu.edu.au/monographs/).
  * [4] Gal, S. G., Niculescu, C. P.: A new look at Popoviciu’s concept for functions of two variables. J. Math. Anal. Appl. 479, 903–925 (2019).
  * [5] Gavrea, B.: On a convexity problem in connection with some linear operators. J. Math. Anal. Appl. 461, 319–332 (2018).
  * [6] Gavrea, B., Gavrea, I.: An Inequality Involving Some Linear Positive Operators and Box-Convex Functions. Results Math. 77(1), 33–43, (2022).
  * [7] Gavrea, B., Gavrea, I.: An Inequality Involving Bernstein Polynomials and Box- Convex Functions. Mediterr. J. Math. 19, 18–33 (2022).
  * [8] Hopf, E.: Uber die Zusammenhange zwischen gewissen hoheren Differenzen-quotienten reelen Funktionen einer reelen Variablen und deren Differenzierbarkeitseigenschaften. Thesis Univ. of Berlin, Berlin, (1926).
  * [9] Komisarski, A., Rajba, T.: Muirhead inequality for convex orders and a problem of I. Raşa on Bernstein polynomials. J. Math. Anal. Appl. 458, 821–830 (2018).
  * [10] Komisarski A., Rajba T.: On the Raşa inequality for higher order convex functions. Results Math. 76(2), 103–115 (2021).
  * [11] Komisarski A., Rajba T.: On the Raşa inequality for higher order convex functions II. Results Math. 77(2), 88–100 (2022).
  * [12] Komisarski A., Rajba T.: On the integral representation and the Raşa, Jensen and Hermite-Hadamard inequalities for box-convex functions. Results Math. 78(3), 89–119 (2023).
  * [13] Mrowiec J., Rajba T., Wa̧sowicz S.: A solution to the problem of Raşa connected with Bernstein polynomials. J. Math. Anal. Appl. 446, 864–878 (2017).
  * [14] Popoviciu T.: Sur quelques proprietes des fonctions d’une ou de deux variables reelles. Mathematica. 8, 1–85 (1934).
  * [15] Popoviciu T.: Les Fonctions Convexes. Hermann. Paris. (1944).
  * [16] Rajba T.: On the Ohlin lemma for Hermite-Hadamard-Fejér type inequalities. Math. Inequal. Appl., 17(2), 557–571 (2014).