#  $L_{p}$-Steiner quermassintegrals 111Keywords: Steiner formula, curvature
measures, (Dual) Brunn Minkowski theory, $L_{p}$ Brunn Minkowski theory, 2020
Mathematics Subject Classification: 52A39 (Primary), 28A75, 52A20, 53A07
(Secondary)

Kateryna Tatarko and Elisabeth M. Werner Supported by Canadian Natural
Sciences and Engineering Research Council (NSERC) Discovery Grant number
2022-02961Partially supported by NSF grant DMS-2103482

###### Abstract

Inspired by an $L_{p}$ Steiner formula for the $L_{p}$ affine surface area
proved by Tatarko and Werner, we define, in analogy to the classical Steiner
formula, $L_{p}$-Steiner quermassintegrals. Special cases include the
classical mixed volumes, the dual mixed volumes, the $L_{p}$ affine surface
areas and the mixed $L_{p}$ affine surface areas. We investigate the
properties of the $L_{p}$-Steiner quermassintegrals in a special class of
convex bodies. In particular, we show that they are rotation and reflection
invariant valuations in this class of convex bodies with a certain degree of
homogeneity. Such valuations seem new and have not been observed before.

##  1 Introduction and Results

An extension of the classical Brunn Minkowski theory, the $L_{p}$ Brunn
Minkowski theory, was initiated by Lutwak in the groundbreaking paper [20].
This theory evolved rapidly over the last years and due to a number of highly
influential works, see, e.g., [5] \- [9], [21] \- [28], [30], [39] \- [41], it
is now a focal part of modern convex geometry. Central objects in the $L_{p}$
Brunn Minkowski theory are the $L_{p}$ affine surface areas,

$as_{p}(K)=\int_{\partial K}\frac{H_{n-1}(x)^{\frac{p}{n+p}}}{\langle
x,N(x)\rangle^{\frac{n(p-1)}{n+p}}}d\mathcal{H}^{n-1}(x),$

where $N(x)$ denotes the outer unit normal at $x\in\partial K$, the boundary
of $K$, $H_{n-1}(x)$ is the Gauss curvature at $x$ and $\mathcal{H}^{n-1}$ is
the usual surface area measure on $\partial K$. A missing part in the $L_{p}$
Brunn Minkowski theory was an analog to the classical Steiner formula [4, 29]
which says that the volume of the outer parallel body $K+tB^{n}_{2}$ of the
convex body $K$ with the Euclidean unit ball $B^{n}_{2}$ is a polynomial in
$t$,

${\rm vol}_{n}(K+tB^{n}_{2})=\sum\limits_{i=0}^{n}{n\choose i}W_{i}(K)t^{i}.$
(1)

The coefficients $W_{i}(K)$ are the classical quermassintegrals. If the
boundary $\partial K$ of the body $K$ is sufficiently smooth, and hence the
principal curvatures are well-defined, then
$W_{i}(K)=\frac{1}{n}\int_{\partial K}H_{i-1}d\mathcal{H}^{n-1}$, where for
$j=1,\dots,n-1$ $H_{j}$ denotes the $j$-th normalized elementary symmetric
function of the principal curvatures, see (4).

In [35] an $L_{p}$ Steiner formula was proved for the $L_{p}$ affine surface
area. Namely, if $K$ is $C^{2}_{+}$, then we have for all suitable $t$ and for
all $p\in\mathbb{R}$, $p\neq-n$, that

$\displaystyle
as_{p}(K+tB^{n}_{2})=\sum\limits_{k=0}^{\infty}\left[\sum\limits_{m=0}^{k}\binom{\frac{n(1-p)}{n+p}}{{k-m}}\mathcal{W}^{p}_{{m},k}(K)\right]t^{k}.$
(2)

The coefficients $\mathcal{W}^{p}_{{m},k}(K)$ are called $L_{p}$ Steiner
coefficients and defined for a (general) convex body $K$ in $\mathbb{R}^{n}$,
for all $k,m\in\mathbb{N}\cup\\{0\\}$ as

$\displaystyle\hskip-31.29802pt\mathcal{W}^{p}_{m,\,k}(K)=$
$\displaystyle\hskip-31.29802pt\int\limits_{\partial K}\langle
x,N(x)\rangle^{m-k+\frac{n(1-p)}{n+p}}H_{n-1}^{\frac{p}{n+p}}\sum_{\begin{subarray}{c}i_{1},\dots,i_{n-1}\geq
0\\\
i_{1}+2i_{2}+\dots+(n-1)i_{n-1}=m\end{subarray}}c(n,p,i(m))\prod\limits_{j=1}^{n-1}{n-1\choose
j}^{i_{j}}H_{j}^{i_{j}}\,d\mathcal{H}^{n-1}.$

The $c(n,p,i(m))$ are certain binomial coefficients, see (18) for the details.
The $\mathcal{W}^{p}_{m,\,k}$ involve integration on $\partial K$. We also
define corresponding expressions $\mathcal{Z}^{p}_{m,\,k}$ in (3.2) as
integrals over the sphere. If $K$ is sufficiently smooth, we can pass from one
expression to the other via the Gauss map. It was also shown in [35] that for
$p=0$ the above formula gives the classical Steiner formula (1). In
particular, the sum appearing in (2) is then a finite sum. The sum is also
finite for all $p=-\frac{n(l-1)}{l}$, $l\in\mathbb{N}$, [35]. In general, the
sum is infinite and thus proving (2) requires a careful analysis of
convergence issues. Moreover, (2) also includes the Steiner formula for the
Minkowski outer parallel body of the dual Brunn Minkowski theory of Lutwak
[17] as a special case. In this paper we define in analogy to the classical
Steiner formula, for a general convex body $K$ in $\mathbb{R}^{n}$ the
$L_{p}$-Steiner quermassintegrals, or $L_{p}$-Steiner mixed volumes as

$\mathcal{V}^{p}_{k}(K)=\sum\limits_{m=0}^{k}\binom{\frac{n(1-p)}{n+p}}{{k-m}}\mathcal{W}^{p}_{{m},k}(K).$
(3)

If $p=0$, then the $L_{p}$-Steiner quermassintegrals (3) are the above
classical quermassintegrals $W_{i}(K)$ (see [4, 29]) and if $p=\pm\infty$,
they are Lutwak’s dual quermassintegrals [17, 19]. Moreover, the $L_{p}$ mixed
affine surface areas of Lutwak [18], see also [39], are special cases of the
$L_{p}$-Steiner quermassintegrals. The $L_{p}$ Steiner coefficients
$\mathcal{W}^{p}_{m,\,k}$ and the $L_{p}$-Steiner quermassintegrals
$\mathcal{V}^{p}_{k}$ are complicated expressions, and one needs to clarify
when they are well-defined. In Section 3 we introduce a class of convex
bodies, the class $\mathcal{C}^{n}$, for which this is the case. We then
investigate in detail the properties of the $L_{p}$-Steiner quermassintegrals.
Remarkably, it turns out that they are new valuations, which seem not to have
been observed before. Valuations on convex sets can be considered as a
generalization of the notion of a measure and thus are important quantities in
the study of convex sets. Examples of valuations which are not measures in the
usual sense are the quermassintegrals and the $L_{p}$ affine surface areas.
Due to their importance, much work has been devoted to the study of
valuations. Often additional properties like continuity with respect to the
Hausdorff metric and invariance under e.g., rigid motions are prescribed. Such
valuations have been completely classified in a remarkable theorem by Hadwiger
as linear combinations of the quermassintegrals in [10]. For a simpler proof,
see [12]. Subsequently, one direction of study in the theory of valuations
concentrated on characterization theorems when one successively relaxes the
requirements. For instance, upper semi continuous, $GL(n)$-invariant
valuations have been characterized by Ludwig and Reitzner [16], involving, in
particular, affine surface area. The following theorem combines some of our
main results, namely (parts of) Theorems 5.1 and 5.5 below. The class
$\mathcal{C}^{n}$ is defined in Section 3. Theorem

Let $K$ be a convex body in $\mathcal{C}^{n}$ and let $p\in\mathbb{R}$ be such
that $p\geq 0$ or $p<-n$. . Then we have for all $k\in\mathbb{N}$

  * (i)

$\mathcal{V}^{p}_{k}$ is homogeneous of degree $n\frac{n-p}{n+p}-k$.

  * (ii)

$\mathcal{V}^{p}_{k}$ is invariant under rotations and reflections.

  * (iii)

$\mathcal{V}^{p}_{k}$ are valuations on the set $\mathcal{C}^{n}$.

In the next two corollaries we list a few consequences of our main theorem.
When $p=1$, the expressions $\mathcal{V}^{p}_{k}$ simplify and we denote
$\mathcal{V}^{1}_{k}=\mathcal{W}^{1}_{k,\,k}=\mathcal{W}_{k,\,k}$.
Additionally to the rotation and reflection invariance, one has translation
invariance for $\mathcal{W}_{k,\,k}$. Corollary

For all $k\in\mathbb{N}$, $\mathcal{V}^{1}_{k}=\mathcal{W}_{k,\,k}$ are rigid
motion invariant, $n\frac{n-1}{n+1}-k$ homogeneous valuations.  For $m=0$, the
$\mathcal{W}^{p}_{{m},k}$ are in addition upper or lower semi continuous
(depending on the range of $p$). As these coincide with the mixed $L_{p}$
affine surface areas $as_{p+\frac{k}{n}(n+p),-k}$, this shows continuity
properties of the latter, which, as far as we know, had not been noted before.
Corollary

For $p\geq 0$, $\mathcal{W}^{p}_{0,\,k}=as_{p+\frac{k}{n}(n+p),-k}$ are upper
semi continuous $n\frac{n-p}{n+p}-k$ homogeneous valuations that are invariant
under rotations and reflections.  In view of these corollaries, we remark that
a characterization of rigid motion invariant upper semi continuous valuations
has been given so far only in the plane [13]. Because of the last corollary
and the fact that the $L_{p}$ affine surface areas are semi continuous, it is
natural to ask about continuity properties of the $\mathcal{V}^{p}_{k}$ and
$\mathcal{W}^{p}_{{m},k}$. We show in Section 5.3 that in general we cannot
expect any semi continuity properties.

A byproduct of our analysis is following combinatorial formula. It may be
known, but we did not find a reference for it. Corollary

Let $p\in\mathbb{R}$, $p\neq-n$ and $k\in\mathbb{N}\cup\\{0\\}$. Then

${n\frac{n-p}{n+p}\choose
k}=\sum\limits_{m=0}^{k}\binom{\frac{n(1-p)}{n+p}}{k-m}\sum_{\begin{subarray}{c}i_{1},\dots,i_{n-1}\geq
0\\\
i_{1}+2i_{2}+\dots+(n-1)i_{n-1}=m\end{subarray}}c(n,p,i(m))\prod\limits_{j=1}^{n-1}{n-1\choose
j}^{i_{j}}$

where $c(n,p,i(m))$ are certain binomial coefficients, see (18) for the
details.

##  2 Background

###  2.1 Background from differential geometry

For more information and the details in this section we refer to e.g., [4,
29].

For a point $x$ on the boundary $\partial K$ of $K$ we denote by $N(x)$ an
outward unit normal vector of $K$ at $x.$ Occasionally, we use $N_{K}(x)$ to
emphasize that it is the normal vector of a body $K$ at $x\in\partial K$. The
map $N:\partial K\to S^{n-1}$ is called the spherical image map or Gauss map
of $K$ and it is of class $C^{1}.$ Its differential is called the Weingarten
map. The eigenvalues of the Weingarten map are the principal curvatures
$k_{i}(x)$ of $K$ at $x.$

The $j$-th normalized elementary symmetric functions of the principal
curvatures are denoted by $H_{j}$. They are defined as follows

$H_{j}={n-1\choose j}^{-1}\sum_{1\leq i_{1}<\dots<i_{j}\leq
n-1}k_{i_{1}}\cdots k_{i_{j}}$ (4)

for $j=1,\dots,n-1$ and $H_{0}=1.$ Note that

$H_{1}=\frac{1}{n-1}\sum_{1\leq i\leq n-1}k_{i}$

is the mean curvature, that is the average of principal curvatures, and

$H_{n-1}=\prod_{i=1}^{n-1}k_{i}$

is the Gauss curvature.

We say that $K$ is of class $C^{2}_{+}$ if $K$ is of class $C^{2}$ and the
Gauss map $N$ is a diffeomorphism. This means in particular that $N$ has a
smooth inverse. This is stronger than just $C^{2},$ and is equivalent to the
assumption that all principal curvatures are strictly positive, or that the
Gauss curvature $H_{n-1}>0.$ It also means that the differential of $N$, i.e.,
the Weingarten map, is of maximal rank everywhere.

Let $K$ be of class $C^{2}_{+}$. For $u\in\mathbb{R}^{n}\setminus\\{0\\}$, let
$\xi_{K}(u)$ be the unique point on the boundary of $K$ at which $u$ is an
outward normal vector. The map $\xi_{K}$ is defined on
$\mathbb{R}^{n}\setminus\\{0\\}$. Its restriction to the sphere $S^{n-1}$ is
called the reverse spherical image map, or reverse Gauss map,
$N_{K}^{-1}:S^{n-1}\to\partial K$. The differential of $N_{K}^{-1}$ is called
the reverse Weingarten map. The eigenvalues of the reverse Weingarten map are
called the principal radii of curvature $r_{1},\dots,r_{n-1}$ of $K$ at $u\in
S^{n-1}.$

The $j$-th normalized elementary symmetric functions of the principal radii of
curvature are denoted by $s_{j}$. In particular, $s_{0}=1$, and for $1\leq
j\leq n-1$ they are defined as

$s_{j}={n-1\choose j}^{-1}\sum_{1\leq i_{1}<\dots<i_{j}\leq
n-1}r_{i_{1}}\cdots r_{i_{j}}.$ (5)

Note that the principal curvatures are functions on the boundary of $K$ and
the principal radii of curvature are functions on the sphere.

Now we describe the connection between $H_{j}$ and $s_{j}.$ For a body $K$ of
class $C^{2}_{+}$, the principal radii of curvature are reciprocals of the
principal curvatures, that is

$r_{i}(u)=\frac{1}{k_{i}(N_{K}^{-1}(u))}.$

This implies that for $x\in\partial K$ with $N(x)=u$,

$s_{j}={n-1\choose j}^{-1}\sum_{1\leq i_{1}<\dots<i_{j}\leq
n-1}\frac{1}{k_{i_{1}}(N_{K}^{-1}(u))\cdots
k_{i_{j}}(N_{K}^{-1}(u))}=\frac{H_{n-1-j}}{H_{n-1}}\Big{(}N_{K}^{-1}(u)\Big{)}$

and

$H_{j}=\frac{s_{n-1-j}}{s_{n-1}}\Big{(}N(x)\Big{)},$

for $j=1,\dots,n-1.$

The mixed volumes $W_{i}(K)$ of the classical Steiner formula (1) can be
expressed via the elementary symmetric functions of the principal curvatures.
By definition, $W_{0}(K)={\rm vol}_{n}(K)$ and

$W_{i}(K)=\frac{1}{n}\int\limits_{\partial K}H_{i-1}d\mathcal{H}^{n-1},$ (6)

for $i=1,\dots,n.$ The dual mixed volume of the convex bodies $K$ and $L$ that
contain $0$ in their interiors, introduced by Lutwak [17], is defined for all
real $i$ by

$\widetilde{V}_{i}(K,L)=\frac{1}{n}\
\int\limits_{S^{n-1}}\rho_{K}(u)^{n-i}\rho_{L}(u)^{i}d\sigma(u),$

where $\rho_{K}(u)=\max\\{\lambda\geq 0|\,\lambda u\in K\\}$ is the radial
function of $K.$ In particular, if $L=B^{n}_{2}$, then

$\widetilde{W}_{i}(K)=\widetilde{V}_{i}(K,B^{n}_{2})=\frac{1}{n}\int\limits_{S^{n-1}}\rho_{K}(u)^{n-i}d\sigma(u)$
(7)

are called dual quermassintegrals of order $i.$ The corresponding Steiner
formula in the dual Brunn Minkowski theory proved in [17] is

${\rm vol}_{n}(K\,\widetilde{+}\,tB^{n}_{2})=\sum\limits_{i=0}^{n}{n\choose
i}\widetilde{W}_{i}(K)\ t^{i}.$

###  2.2 Background from affine geometry

We denote by $\mathcal{K}^{n}_{o}$ the set of all convex bodies in
$\mathbb{R}^{n}$ containing the origin $o$. From now on, we will always assume
that $K\in\mathcal{K}^{n}_{o}$. For real $p\neq-n$, we define the $L_{p}$
affine surface area $as_{p}(K)$ of $K$ as in [17] ($p>1$) and [33] ($p<1,\
p\neq-n$) by

$as_{p}(K)=\int\limits_{\partial K}\frac{H_{n-1}(x)^{\frac{p}{n+p}}}{\langle
x,N(x)\rangle^{\frac{n(p-1)}{n+p}}}\,d\mathcal{H}^{n-1}(x)$ (8)

and

$as_{\pm\infty}(K)=\int\limits_{\partial K}\frac{H_{n-1}(x)}{\langle
x,N(x)\rangle^{n}}d\mathcal{H}^{n-1}(x)$ (9)

provided the above integrals exist. Note that these expressions are not always
finite. For example, if $K$ is a polytope and $-n<p<0$, then
$as_{p}(K)=\infty$.

In particular, for $p=0$,

$as_{0}(K)=\int\limits_{\partial K}\langle
x,N(x)\rangle\,d\mathcal{H}^{n-1}(x)=n\,{\rm vol}_{n}(K).$ (10)

The case $p=1$,

$as_{1}(K)=\int\limits_{\partial
K}H_{n-1}(x)^{\frac{1}{n+1}}d\mathcal{H}^{n-1}(x)$

is the classical affine surface area which is independent of the position of
$K$ in space. For dimensions 2 and 3 and sufficiently smooth convex bodies,
its definition goes back to Blaschke [2].

If the boundary of $K$ is sufficiently smooth, then (8) and (9) can be written
as integrals over the boundary $\partial B^{n}_{2}=S^{n-1}$ of the Euclidean
unit ball $B^{n}_{2}$ in $\mathbb{R}^{n}$,

$as_{p}(K)=\int\limits_{S^{n-1}}\frac{f_{K}(u)^{\frac{n}{n+p}}}{h_{K}(u)^{\frac{n(p-1)}{n+p}}}d\mathcal{H}^{n-1}(u),$
(11)

where $f_{K}(u)$ is the curvature function, i.e. the reciprocal of the
Gaussian curvature $H_{n-1}(x)$ at this point $x\in\partial K$ that has $u$ as
outer normal, and $h_{K}(u)=\sup\\{\langle x,u\rangle:x\in K\\}$,
$u\in\mathbb{R}^{n}\setminus\\{0\\}$, is the support function of $K$. In
particular, for $p=\pm\infty$,

$as_{\pm\infty}(K)=\int\limits_{S^{n-1}}\frac{1}{h_{K}(u)^{n}}d\mathcal{H}^{n-1}(x)=n\,{\rm
vol}_{n}(K^{\circ}),$ (12)

where $K^{\circ}=\\{y\in\mathbb{R}^{n}:\langle x,y\rangle\leq 1\ \text{for all
}x\in K\\}$ is the polar body of $K$. For $p=-n$, the $L_{-n}$ affine surface
area was introduced in [26] as

$as_{-n}(K)=\max_{u\in
S^{n-1}}f_{K}(u)^{\frac{1}{2}}h_{K}(u)^{\frac{n+1}{2}}.$ (13)

Hug [11] proved that (8) and (11) coincide for $p>0$.

##  3 $L_{p}$ Steiner quermassintegrals

###  3.1 Definitions

We will need the generalized binomial coefficients. For $\alpha\in\mathbb{R}$
and $k\in\mathbb{N}$, they are defined as

${\alpha\choose k}=\left\\{\begin{aligned} &1\quad&\text{ if }\ k=0,\\\
&0\quad&\text{ if }\,k<0\,&\text{ or }\,\alpha=0,\\\
&\frac{\alpha(\alpha-1)\cdots(\alpha-k+1)}{k!}\quad&\text{ if
}k>0.\end{aligned}\right.$ (14)

It was shown in [35] that the following $L_{p}$ Steiner formulas hold. They
are a generalization of the classical Steiner formula, which corresponds to
the case $p=0$ and Lutwak’s Steiner formula of the dual Brunn Minkowski
theory, which corresponds to the case $p=\pm\infty$:

###### Theorem 3.1

[35] Let $k,m\in\mathbb{N}\cup\\{0\\}$, and $p\in\mathbb{R}$, $p\neq-n$. If
$K\in\mathcal{K}_{o}^{n}$ is $C^{2}_{+}$, then we have for all $0\leq
t\leq\min\limits_{u\in S^{n-1}}h_{K}(u)$,

$\displaystyle
as_{p}(K+tB^{n}_{2})=\sum\limits_{k=0}^{\infty}\left[\sum\limits_{m=0}^{k}\binom{\frac{n(1-p)}{n+p}}{{k-m}}\mathcal{W}^{p}_{{m},k}(K)\right]t^{k}=\sum\limits_{k=0}^{\infty}\left[\sum\limits_{m=0}^{k}\binom{\frac{n(1-p)}{n+p}}{k-m}\mathcal{Z}^{p}_{m,\,k}(K)\right]t^{k}.$
(15)

A convex body $K$ has a unique outer normal almost everywhere, and by a
theorem of Alexandroff [1] and Busemann-Feller [3] the generalized second
partial derivatives exist almost everywhere. Therefore, we extend the
definition of the $L_{p}$ Steiner coefficients from smooth convex bodies [35]
to general convex bodies $K$.

######  Definition 3.2 ($L_{p}$-Steiner coefficients)

Let $K\in\mathcal{K}_{o}^{n}$, $k,m\in\mathbb{N}\cup\\{0\\}$, and
$p\in\mathbb{R}$, $p\neq-n$. Then we define

$\displaystyle\hskip-31.29802pt\,\,\mathcal{W}^{p}_{m,\,k}(K)=$
$\displaystyle\hskip-31.29802pt\int\limits_{\partial K}\langle
x,N(x)\rangle^{m-k+\frac{n(1-p)}{n+p}}H_{n-1}^{\frac{p}{n+p}}\sum_{\begin{subarray}{c}i_{1},\dots,i_{n-1}\geq
0\\\
i_{1}+2i_{2}+\dots+(n-1)i_{n-1}=m\end{subarray}}c(n,p,i(m))\prod\limits_{j=1}^{n-1}{n-1\choose
j}^{i_{j}}H_{j}^{i_{j}}\,d\mathcal{H}^{n-1}$ (16)

and

$\displaystyle\hskip-31.29802pt\,\,\mathcal{Z}^{p}_{m,\,k}(K)=$
$\displaystyle\hskip-31.29802pt\int\limits_{S^{n-1}}h_{K}^{m-k+\frac{n(1-p)}{n+p}}\,s_{n-1}^{\frac{n}{n+p}}(u)\sum_{\begin{subarray}{c}i_{1},\dots,i_{n-1}\geq
0\\\
i_{1}+2i_{2}+\dots+(n-1)i_{n-1}=m\end{subarray}}c(n,p,i(m))\frac{\prod\limits_{j=1}^{n-1}{n-1\choose
j}^{i_{j}}s_{n-1-j}^{i_{j}}\left(u\right)}{s_{n-1}^{\sum\limits_{j}i_{j}}(u)}\,d\mathcal{H}^{n-1}.$
(17)

There we have put

$c(n,p,i(m))={\frac{n}{n+p}\choose
i_{1}+\dots+i_{n-1}}{i_{1}+\dots+i_{n-1}\choose
i_{1},\,i_{2},\,\ldots,i_{n-1}},$ (18)

where the sequence $i(m)=\\{i_{j}\\}_{j=1}^{n-1}$ is such that
$i_{1}+2i_{2}+\dots+(n-1)i_{n-1}=m$, and

${q\choose i_{1},i_{2},\dots,i_{l}}=\frac{q!}{i_{1}!i_{2}!\cdots i_{l}!}$

is the multinomial coefficient where $q=i_{1}+\dots+i_{l}$. Note that

${q\choose i_{1},i_{2},\dots,i_{l}}=0\quad\text{ if }\ i_{j}<0\text{ or
}i_{j}>q.$

When $l=2,$ we get the binomial coefficients.

Remark (see [35]) If $p=1$, the $L_{p}$ Steiner formula (15) reduces to

$\displaystyle as_{1}(K+tB^{n}_{2})$ $\displaystyle=$
$\displaystyle\sum\limits_{k=0}^{\infty}\mathcal{W}_{k,\,k}(K)\,t^{k}=\sum\limits_{k=0}^{\infty}\mathcal{Z}_{k,\,k}(K)\,t^{k},$
(19)

and the expressions (3.2) and (3.2) simplify to

$\displaystyle 1.\,\,\mathcal{W}_{k,\,k}(K)$ $\displaystyle=$
$\displaystyle\mathcal{W}^{1}_{k,\,k}(K)=$ (20) $\displaystyle=$
$\displaystyle\int\limits_{\partial
K}H_{n-1}^{\frac{1}{n+1}}\sum_{\begin{subarray}{c}i_{1},\dots,i_{n-1}\geq 0\\\
i_{1}+2i_{2}+\dots+(n-1)i_{n-1}=k\end{subarray}}c(n,1,i(k))\prod\limits_{j=1}^{n-1}{n-1\choose
j}^{i_{j}}H_{j}^{i_{j}}\left(x\right)\,d\mathcal{H}^{n-1}$

and

$\displaystyle 2.\,\,\mathcal{Z}_{k,\,k}(K)$ $\displaystyle=$
$\displaystyle\mathcal{Z}^{1}_{k,\,k}(K)=$ (21) $\displaystyle=$
$\displaystyle\int\limits_{S^{n-1}}s_{n-1}^{\frac{n}{n+1}}\sum_{\begin{subarray}{c}i_{1},\dots,i_{n-1}\geq
0\\\
i_{1}+2i_{2}+\dots+(n-1)i_{n-1}=k\end{subarray}}c(n,1,i(k))\frac{\prod\limits_{j=1}^{n-1}{n-1\choose
j}^{i_{j}}s_{n-1-j}^{i_{j}}}{s_{n-1}^{\sum\limits_{j}i_{j}}(u)}\,d\mathcal{H}^{n-1}.$

The natural question is for which class of bodies and for which parameters the
quantities (3.2) and (3.2) are well-defined. We are investigating this next.
To do so, we introduce the notions of inner and outer rolling balls.

We say that $K$ admits an inner rolling ball of radius $r>0$ if for any
$x\in\partial K$

$B^{n}_{2}(x-rN(x),r)\subseteq K,$

and $K$ admits an outer rolling ball of radius $R>0$ if for any $x\in\partial
K$

$K\subseteq B^{n}_{2}(x-RN(x),R).$

We then define the following class of convex bodies:

$\displaystyle\mathcal{C}^{n}=\\{K\in\mathcal{K}_{o}^{n}:\ K\ \textup{admits
an inner and outer rolling ball}\\}.$ (22)

The class $\mathcal{C}^{n}$ proved useful in many contexts, e.g. in
approximation of convex bodies by random polytopes [32]. Note that if $K\
\in\mathcal{C}^{n}$ then $K$ is $C^{1}$ and strictly convex. Moreover, if $K$
is $C^{2}_{+}$ then $K$ is in $\mathcal{C}^{n}$ but the converse does not hold
in general. We discuss the class $\mathcal{C}^{n}$ in detail in Section 4. In
particular, we show that the $L_{p}$ Steiner coefficients (for
$p\in\mathbb{R}$ such that $\frac{p}{n+p}\geq 0$) are finite for any
$K\in\mathcal{C}^{n}$. As $K\in\mathcal{C}^{n}$ is $C^{1}$ we can pass from
$\mathcal{W}^{p}_{m,k}$ to $\mathcal{Z}^{p}_{m,\,k}$ via the Gauss map
$N:\partial K\to S^{n-1}$, and as $K$ is strictly convex, we can pass from
$\mathcal{Z}^{p}_{m,\,k}$ to $\mathcal{W}^{p}_{m,k}$ via the inverse of the
Gauss map. Thus, it is enough to consider $\mathcal{W}^{p}_{m,k}$.

In analogy to the classical Steiner formula (1), Theorem 3.1 leads us to
define the $L_{p}$-Steiner quermassintegrals.

######  Definition 3.3 ($L_{p}$-Steiner quermassintegrals)

Let $K$ be a convex body in $\mathbb{R}^{n}$, $p\in\mathbb{R}$, $p\neq-n$ and
$k\in\mathbb{N}\cup\\{0\\}$. Then we define the $L_{p}$-Steiner
quermassintegrals

$\mathcal{V}_{k}^{p}(K)=\sum\limits_{m=0}^{k}\binom{\frac{n(1-p)}{n+p}}{k-m}\mathcal{W}^{p}_{m,\,k}(K),$
(23)

and

$\mathcal{U}_{k}^{p}(K)=\sum\limits_{m=0}^{k}\binom{\frac{n(1-p)}{n+p}}{k-m}\mathcal{Z}^{p}_{m,\,k}(K).$
(24)

We will mainly consider convex bodies that are in the class $\mathcal{C}^{n}$
and $p\in\mathbb{R}$ such that $\frac{p}{n+p}\geq 0$. Then we can pass from
$\mathcal{U}^{p}_{k}$ to $\mathcal{V}^{p}_{k}$ via the Gauss map $N:\partial
K\to S^{n-1}$ and as $K$ is strictly convex, we can pass from
$\mathcal{V}^{p}_{k}$ to $\mathcal{U}^{p}_{k}$ via the inverse of the Gauss
map. Thus, it suffices to consider $\mathcal{V}^{p}_{k}$. If $p=1$, then

$\mathcal{V}_{k}^{1}(K)=\mathcal{W}_{k,\,k}(K)\hskip
28.45274pt\text{and}\hskip
28.45274pt\mathcal{U}_{k}^{1}(K)=\mathcal{Z}_{k,\,k}(K).$

In general, the expressions $\mathcal{V}_{k}^{p}(K)$ may become infinite or
undetermined, depending on the body $K$ and the parameter $p$. In Section 4,
we will discuss this issue and we will show that $\mathcal{V}_{k}^{p}(K)$ are
finite for $K\in\mathcal{C}^{n}$ and $p\in\mathbb{R}$ such that
$\frac{p}{n+p}\geq 0$.

Remark 1\. Note that the $L_{p}$-Steiner quermassintegrals as well as $L_{p}$
Steiner coefficients can be negative. This is not the case for the classical
quermassintegrals.

2\. Nevertheless, the $L_{p}$-Steiner quermassintegrals closely parallel the
classical Steiner coefficients. This is further illustrated next. In the
classical Steiner formula (1), the first and the last coefficients are

$W_{0}(K)={\rm vol}_{n}(K)\,\,\,\text{ and}\,\,\,\,W_{n}(K)={\rm
vol}_{n}(B^{n}_{2}),$

respectively. In the $L_{p}$ Steiner formula (15), the first coefficient is
$\mathcal{V}_{0}^{p}(K)=as_{p}(K)$. It was shown in [35] that when
$p=\frac{-n(l-1)}{l}$, $l\in\mathbb{N}$, the sums (15) are finite with the
highest term $t^{n(2l-1)}$:

$\displaystyle
as_{-\frac{n(l-1)}{l}}(K+tB^{n}_{2})=\sum_{k=0}^{l(n-1)}\left[\sum_{m=0}^{k}\binom{l+n(l-1)}{k-m}\mathcal{W}_{m,\,k}(K)\right]t^{k}\,+$
$\displaystyle\sum_{k=l(n-1)+1}^{l+n(l-1)}\left[\sum_{m=0}^{l(n-1)}\binom{l+n(l-1)}{k-m}\mathcal{W}_{m,\,k}(K)\right]t^{k}+\sum_{k=l+n(l-1)+1}^{n(2l-1)}\left[\sum_{m=k-(l+n(l-1))}^{l(n-1)}\binom{l+n(l-1)}{k-m}\mathcal{W}_{m,\,k}(K)\right]t^{k}.$

Then the last coefficient is

$\displaystyle\int\limits_{S^{n-1}}s_{n-1}^{l}(u)\sum_{\begin{subarray}{c}i_{1},\dots,i_{n-1}\geq
0\\\ i_{1}+\dots+(n-1)i_{n-1}=l(n-1)\end{subarray}}{l\choose
i_{1}+\dots+i_{n-1}}{i_{1}+\dots+i_{n-1}\choose
i_{1},\,\ldots,i_{n-1}}\frac{\prod\limits_{j=1}^{n-1}{n-1\choose
j}^{i_{j}}s_{n-1-j}^{i_{j}}\left(u\right)}{s_{n-1}^{\sum\limits_{j}i_{j}}(u)}\,d\mathcal{H}^{n-1}(u).$

We only get a contribution in the above sum if $i_{1}+\dots+i_{n-1}\leq l$ and
$i_{1}+2i_{2}+\dots+(n-1)i_{n-1}=l(n-1).$ Therefore,

$i_{1}+2i_{2}+\dots+(n-2)i_{n-2}=(l-i_{n-1})(n-1)\geq(n-1)(i_{1}+\dots+i_{n-2}),$

since $i_{1}+\dots+i_{n-1}\leq l.$ This is true only when
$i_{1}=i_{2}=\dots=i_{n-2}=0$. Thus, the only possible choice of indices is
$i_{1}=i_{2}=\dots=i_{n-2}=0$ and $i_{n-1}=l.$ Then the coefficient of
$t^{n(2l-1)}$, i.e. the last coefficient, is

$\displaystyle\int\limits_{S^{n-1}}\,d\mathcal{H}^{n-1}(u)={\rm
vol}_{n-1}(\partial B^{n}_{2})=as_{p}(B^{n}_{2}),$

which shows the parallel behavior of the classical and $L_{p}$ Steiner
coefficients.

##  4 A first analysis of the $L_{p}$-Steiner quermassintegrals

###  4.1 The class $\mathcal{C}^{n}$

###### Proposition 4.1

Let $p\in\mathbb{R}$ be such that $p\geq 0$ or $p<-n$. Let
$k,m\in\mathbb{N}\cup\\{0\\}$ and let the sequence
$i(m)=\\{i_{j}\\}_{j=1}^{n-1}$ be such that
$i_{1}+2i_{2}+\dots+(n-1)i_{n-1}=m$. Let $\mathcal{C}^{n}$ be defined as in
(22). Then we have for any $K\in\mathcal{C}^{n}$

$-\infty<\mathcal{W}^{p}_{m,\,k}(K)<\infty\qquad\textup{and}\qquad-\infty<\mathcal{V}^{p}_{k}(K)<\infty.$

Proof  The $L_{p}$ Steiner coefficients $\mathcal{W}^{p}_{m,\,k}$ (3.2), and
thus the $L_{p}$-Steiner quermassintegrals $\mathcal{V}^{p}_{k}$ (23), are
sums (up to coefficients) of expressions of the form

$\int\limits_{\partial K}\langle
x,N(x)\rangle^{m-k+\frac{n(1-p)}{n+p}}H_{n-1}^{\frac{p}{n+p}}\,\prod\limits_{j=1}^{n-1}{n-1\choose
j}^{i_{j}}H_{j}^{i_{j}}\,d\mathcal{H}^{n-1}.$ (25)

As $K$ is a convex body in $\mathcal{K}_{o}^{n}$, there is $a>0$,
$a\in\mathbb{R}$, such that $B^{n}_{2}(0,a)\subset K\subset
B^{n}_{2}\left(0,\frac{1}{a}\right)$. Thus

$a\leq\langle x,N(x)\rangle\leq\frac{1}{a}.$ (26)

Since $K$ admits an inner rolling ball with radius $r$ and an outer rolling
ball with radius $R$, we have for all $x\in\partial K$

$\frac{1}{R^{n-1}}\leq H_{n-1}(x)\leq\frac{1}{r^{n-1}}$

and each $k_{i}(x)$ is bounded from above and below by $\frac{1}{r}$ and
$\frac{1}{R}$, respectively. This implies

$\displaystyle\gamma\left(\frac{1}{R}\right)^{m+(n-1)\frac{p}{n+p}}{\rm
vol}_{n-1}(\partial K)\leq\int\limits_{\partial
K}H_{n-1}^{\frac{p}{n+p}}\,\prod\limits_{j=1}^{n-1}{n-1\choose
j}^{i_{j}}H_{j}^{i_{j}}\,d\mathcal{H}^{n-1}\leq\gamma\left(\frac{1}{r}\right)^{m+(n-1)\frac{p}{n+p}}{\rm
vol}_{n-1}(\partial K).$

where $\gamma=\prod\limits_{j=1}^{n-1}{n-1\choose j}^{i_{j}}.$

Combining this with (26), gives that the $L_{p}$ Steiner coefficients
$\mathcal{W}^{p}_{m,\,k}$ are finite. Thus the $\mathcal{V}^{p}_{k}$ are also
finite as a finite sum of $\mathcal{W}^{p}_{m,\,k}$.

The next example shows that the assumptions on the parameter $p$ and the
rolling ball assumptions are needed so that the $L_{p}$-Steiner
quermassintegrals are finite.

Example 4.2 Let $1<r<\infty$ and
$B_{r}^{n}=\\{x\in\mathbb{R}^{2}:|x_{1}|^{r}+\cdots+|x_{n}|^{r}\leq 1\\}$. For
$m=0$

$\mathcal{W}^{p}_{0,\,k}(B_{r}^{n})=\int\limits_{\partial K}\langle
x,N(x)\rangle^{-k+\frac{n(1-p)}{n+p}}H_{n-1}^{\frac{p}{n+p}}\,d\mathcal{H}^{n-1}=\int\limits_{\partial
K}\langle x,N(x)\rangle^{\frac{n(1-p)}{n+p}}H_{n-1}^{\frac{p}{n+p}}\langle
x,N(x)\rangle^{-k}\,d\mathcal{H}^{n-1}.$

Using (26), we get

$a^{k}as_{p}(B_{r}^{n})\leq\mathcal{W}^{p}_{0,\,k}(B_{r}^{n})\leq\frac{1}{a^{k}}as_{p}(B_{r}^{n}).$

Now we observe the following:

(i) Let $2<r<\infty$. Then $B^{n}_{r}$ admits inner and outer rolling balls.
It was shown in [33] that for $-n<p<-\frac{n}{r-1}$

$as_{p}(B_{r}^{n})=\infty.$

This shows that the assumption $\frac{p}{n+p}\geq 0$ is needed.

(ii) Let $1<r<2$. Then $B^{n}_{r}$ does not admit an inner rolling ball. It
was shown in [33] that

$as_{p}(B_{r}^{n})=\infty$

for $-\frac{n}{r-1}\leq p<-n$. This case shows that the assumption on rolling
balls is needed.

###### Proposition 4.2

The class $\mathcal{C}^{n}$ is closed under intersections and unions assuming
that the union is convex.

Proof  Let $K$ and $L$ be convex bodies in $\mathcal{C}^{n}$. First, we show
that $K\cap L$ and $K\cup L$ admit an inner rolling ball. We start with $K\cup
L$ and split its boundary into disjoint sets as follows

$\partial(K\cup L)=\\{\partial K\cap\partial L\\}\cup\\{\partial K\cap
L^{c}\\}\cup\\{K^{c}\cap\partial L\\}.$

Then we consider the following cases:

  * 1)

Let $x\in\partial K\cap\partial L.$ Since $x\in\partial K$ and $K$ is in
$\mathcal{C}^{n}$, then $N_{K}(x)$ is unique. At the same time, since
$x\in\partial L$ and $L$ is in $\mathcal{C}^{n}$, then $N_{L}(x)$ is unique.
Hence, $N_{K}(x)=N_{L}(x)=N(x)$ and there are inner rolling balls of $K$ and
$L$ at $x$ with radii $r_{K}$ and $r_{L}$, respectively. We take
$r=\min\\{r_{K},r_{L}\\}$. Then $B^{n}_{2}(x-rN(x),r)\subseteq K\cup L$.

  * 2)

Let $x\in\partial K\cap L^{c}$. Then
$B^{n}_{2}(x-r_{K}N_{K}(x),r_{K})\subseteq K$. By taking
$r=\min\\{r_{K},r_{L}\\}$ and $N(x)=N_{K}(x)$, we get that
$B^{n}_{2}(x-rN(x),r)\subseteq K$ which implies $B^{n}_{2}(x-rN(x),r)\subseteq
K\cup L.$

  * 3)

The proof that there is an inner rolling ball when $x\in K^{c}\cap\partial L$
is similar to the case 2).

Next, we deal with $K\cap L$ and decompose its boundary as

$\partial(K\cap L)=\\{\partial K\cap\partial L\\}\cup\\{\partial
K\cap\operatorname{int}(L)\\}\cup\\{\operatorname{int}(K)\cap\partial L\\}.$

Again, we consider the following cases:

  * 1)

Let $x\in\partial K\cap\partial L.$ Then the proof is similar to the case 1)
for $K\cup L$.

  * 2)

Let $x\in\partial K\cap\operatorname{int}(L)$. The case when
$x\in\operatorname{int}(K)\cap\partial L$ is treated similarly.

Let $z=\\{\lambda x:\lambda\geq 0\\}\cap\partial L$.

    * 2a)

Assume first that $z\in\partial L$ is parallel to $N_{L}(z)$. Consider the
convex cone that is the convex hull of the origin $o$ and the inner ball
$B^{n}_{2}(z-r_{L}N_{L}(z),r_{L})$ (see Figure 1). Observe that this cone is
contained in $L$. Using similar triangles, we get

$\frac{\rho}{r_{L}}=\frac{||x||}{||z||-r_{L}},$

where $||\cdot||$ denotes the Euclidean norm. Without loss of generality, we
can assume that $r_{L}\leq\frac{1}{2}||z||$ for any $z\in\partial L$. If
$r_{L}=||z||$ for some $z$, we choose $r_{L}=\frac{1}{2}||z||$. Then

$\frac{||x||}{||z||}r_{L}\leq\rho=\frac{||x||}{||z||-r_{L}}r_{L}\leq
2\frac{||x||}{||z||}r_{L}.$

We put $\rho_{0}=r_{L}\frac{\min\limits_{x\in\partial
K}||x||}{\max\limits_{z\in\partial L}||z||}$. The set
$B^{n}_{2}(x,\rho_{0})\cap B_{2}^{n}(x-r_{K}N_{K}(x),r_{K})$ has non-empty
interior. Choosing $r=\min\\{\frac{\rho_{0}}{4},\frac{r_{K}}{4}\\}$ gives that

$B_{2}^{n}(x-rN_{K}(x),r)\subseteq B_{2}^{n}(x-r_{K}N_{K}(x),r_{K})\subseteq
K$

and

$B_{2}^{n}(x-rN_{K}(x),r)\subseteq B^{n}_{2}(x,\rho_{0})\subseteq L.$

This implies that $B_{2}^{n}(x-rN_{K}(x),r)\subseteq K\cap L$.

Figure 1:  Case 2a).

    * 2b)

Now we treat the general case, when $z\in\partial L$ is not necessarily
parallel to $N_{L}(z)$. We reduce this case to the previous case. As
$x\in\text{int}(L)$, the line segment

$[z_{0},z]=\\{\lambda x:\lambda\geq 0\\}\cap B_{2}^{n}(z-r_{L}N_{L}(z),r_{L})$

is in the interior of $B_{2}^{n}(z-r_{L}N_{L}(z),r_{L})$. Let $w$ be the
midpoint of $[z_{0},z]$ and let $\alpha={\rm
dist}(w,\partial(B_{2}^{n}(z-r_{L}N_{L}(z),r_{L})))$ be the distance from $w$
to the boundary of $B_{2}^{n}(z-r_{L}N_{L}(z),r_{L}))$. Then

$B_{2}^{n}(w,\alpha/2)\subset B_{2}^{n}(z-r_{L}N_{L}(z),r_{L}).$

Now we consider the cone that is the convex hull of the origin $o$ and the
ball $B^{n}_{2}(w,\alpha/2)$. We then proceed as above, replacing $z$ by
$w+\alpha/2\frac{x}{||x||}$, also noting that
$||w+\alpha/2\frac{x}{||x||}||\leq||z||$.

The proof that $K\cap L$ and $K\cup L$ admit an outer rolling ball is similar
to the above, though slightly more technical.

###  4.2 Special Cases

The $L_{p}$-Steiner quermassintegrals generalize the known quermassintegrals,
the dual mixed volumes and the mixed $L_{p}$ affine surface areas. Indeed, (i)
If $p=0$ and $K$ is $C^{2}_{+}$,

$\mathcal{V}^{0}_{k}(K)=n\binom{n}{k}\,W_{k}(K).$ (27)

Then (27) can be deduced immediately as follows. From (15) we get that

$as_{0}(K+tB^{n}_{2})=\sum_{k=0}^{\infty}\mathcal{V}^{0}_{k}(K)\,t^{k}.$ (28)

On the other hand, by the classical Steiner formula,

$as_{0}(K+tB^{n}_{2})=n\,{\rm
vol}_{n}\left(K+tB^{n}_{2}\right)=\sum_{k=0}^{n}\binom{n}{k}\,\mathcal{W}_{k}(K)\,t^{k}.$
(29)

Comparing (28) and (29), we see that $\mathcal{V}^{0}_{k}(K)=0$ for all $k>n$
and for $0\leq k\leq n$ (27) follows immediately. (ii) If $p=\pm\infty$ and
$K$ is $C^{2}_{+}$, we get from the definition that

$\mathcal{V}^{\pm\infty}_{k}(K)=\binom{-n}{k}\widetilde{W}_{-k}(K^{\circ})=(-1)^{k}\binom{n+k-1}{k}\widetilde{W}_{-k}(K^{\circ}),$

where $\widetilde{W}_{k}(K)$ are the dual quermassintegrals, [17]. (iii) The
$L_{p}$-Steiner quermassintegrals generalize the known mixed $L_{p}$ affine
surface areas [18, 39]. We recall that for all $p\neq-n$ and all real $s$, the
$s$-th mixed $L_{p}$ affine surface area of $K$ is defined as

$as_{p,\,s}(K)=\int\limits_{\partial K}H_{n-1}(x)^{\frac{s+p}{n+p}}\langle
x,N(x)\rangle^{(1-p)\frac{n-s}{n+p}}d\mathcal{H}^{n-1}(x).$

Then we get for $k=l(n-1)$, $l\in\mathbb{N}$ and $p=1$,

$\mathcal{V}_{l(n-1)}^{1}(K)=\mathcal{W}_{l(n-1),\,l(n-1)}(K)=\binom{\frac{n}{n+1}}{l}as_{1,l(n+1)}(K).$

Similarly, for $m=0$,

$\mathcal{W}^{p}_{0,\,k}(K)=\int\limits_{\partial K}\langle
x,N(x)\rangle^{-k+\frac{n(1-p)}{n+p}}H_{n-1}^{\frac{p}{n+p}}\,d\mathcal{H}^{n-1}(x)=as_{p+\frac{k}{n}(n+p),\,-k}(K).$
(30)

Note that the term $\mathcal{W}^{p}_{0,\,k}(K)$, which corresponds to the last
summand in $\mathcal{V}^{p}_{k}(K)$, is finite and positive for all $k$ and
$p$ such that $\frac{p}{n+p}\geq 0$. However, for $p<0$, it may become
infinite (see Example 4.2). When, in addition, $p=0$ and $k=1$,
$\mathcal{W}^{0}_{0,\,1}(K)$ is the surface area ${\rm vol}_{n-1}(\partial K)$
of $K$. (iv) For the Euclidean unit ball $B^{n}_{2}$ and all parameters
$p\neq-n$ we get that

$\displaystyle\mathcal{V}^{p}_{k}(B^{n}_{2})$ $\displaystyle=$
$\displaystyle\sum\limits_{m=0}^{k}\binom{\frac{n(1-p)}{n+p}}{k-m}\mathcal{W}^{p}_{m,\,k}(B^{n}_{2})$
(31) $\displaystyle=$ $\displaystyle{\rm vol}_{n-1}(\partial
B^{n}_{2})\,\sum\limits_{m=0}^{k}\binom{\frac{n(1-p)}{n+p}}{k-m}\sum_{\begin{subarray}{c}i_{1},\dots,i_{n-1}\geq
0\\\
i_{1}+2i_{2}+\dots+(n-1)i_{n-1}=m\end{subarray}}c(n,p,i(m))\prod\limits_{j=1}^{n-1}{n-1\choose
j}^{i_{j}}.$

We define

$C(n,p,k)=\sum\limits_{m=0}^{k}\binom{\frac{n(1-p)}{n+p}}{k-m}\sum_{\begin{subarray}{c}i_{1},\dots,i_{n-1}\geq
0\\\
i_{1}+2i_{2}+\dots+(n-1)i_{n-1}=m\end{subarray}}c(n,p,i(m))\prod\limits_{j=1}^{n-1}{n-1\choose
j}^{i_{j}}.$ (32)

Then,

$\mathcal{V}^{p}_{k}(B^{n}_{2})=C(n,p,k)\,{\rm vol}_{n-1}(\partial
B^{n}_{2}).$ (33)

In particular,

$C(n,1,k)=\sum_{\begin{subarray}{c}i_{1},\dots,i_{n-1}\geq 0\\\
i_{1}+2i_{2}+\dots+(n-1)i_{n-1}=k\end{subarray}}c(n,1,i(k))\prod\limits_{j=1}^{n-1}{n-1\choose
j}^{i_{j}}$ (34)

and

$\mathcal{W}_{k,\,k}(B^{n}_{2})={\rm vol}_{n-1}(\partial
B^{n}_{2})\,C(n,1,k).$ (35)

(v) Polytopes $P$ are not in $\mathcal{C}_{p}$ but we would like to note that
for polytopes the $L_{p}$-Steiner quermassintegrals exhibit similar properties
as the $L_{p}$ affine surface areas. By (30),

$\mathcal{W}_{0,\,k}^{0}(P)=\int\limits_{\partial P}\langle
x,N(x)\rangle^{-k+1}\,d\mathcal{H}^{n-1}(x)=as_{k,\,-k}(P),$

and by definition,

$\mathcal{V}^{0}_{k}(P)=\binom{1}{k}\,\mathcal{W}_{0,\,k}^{0}(P)=\left\\{\begin{aligned}
&n\,{\rm vol}_{n}(P)\quad&\text{ if }\,k=0\\\ &{\rm vol}_{n-1}(\partial
P)\quad&\text{ if }\,k=1\\\ &0\quad&\text{ if }\,\,\,k\geq
2.\end{aligned}\right.$ (36)

For $p\neq 0$, we get for all $k\geq 0$,

$\mathcal{V}^{p}_{k}(P)=0\,\,\text{ if }\,\,0<p\leq\infty\,\text{ and
}\,-\infty\leq p<-n.$ (37)

###  4.3 The expressions $C(n,p,k)$

The $L_{p}$ affine surface area is homogeneous of degree $n\frac{n-p}{n+p}$,
i.e.,

$as_{p}(tK)=t^{n\frac{n-p}{n+p}}as_{p}(K).$ (38)

Hence

$as_{p}((1+t)B^{n}_{2})=(1+t)^{n\frac{n-p}{n+p}}{\rm vol}_{n-1}(\partial
B^{n}_{2})={\rm vol}_{n-1}(\partial
B^{n}_{2})\sum_{k=0}^{\infty}{n\frac{n-p}{n+p}\choose k}\ t^{k}.$ (39)

Observe that for $p=n$, $as_{n}((1+t)B^{n}_{2})={\rm vol}_{n-1}(\partial
B^{n}_{2})$ and that on the right hand side of (39) ${0\choose k}=0$ for all
$k$, except for $k=0$, when ${0\choose 0}=1$. In particular, this means that

$C(n,n,k)=\left\\{\begin{aligned} &1\quad&\text{ if }\ k=0,\\\ &0\quad&\text{
if }\,k\geq 1.\end{aligned}\right.$

For $p\neq n,-n$, applying (15) when $K$ is the Euclidean unit ball
$B^{n}_{2}$, we get

$as_{p}(B^{n}_{2}+tB^{n}_{2})=\sum_{k=0}^{\infty}\mathcal{V}_{k}^{p}(B^{n}_{2})\,t^{k}.$

Therefore,

$\mathcal{V}_{k}^{p}(B^{n}_{2})={\rm vol}_{n-1}(\partial
B^{n}_{2}){n\frac{n-p}{n+p}\choose k}$

which implies that

$C(n,p,k)={n\frac{n-p}{n+p}\choose k}.$ (40)

This observation results in the following combinatorial formula.

###### Corollary 4.3

Let $p\in\mathbb{R}$, $p\neq-n$. Then

${n\frac{n-p}{n+p}\choose
k}=\sum\limits_{m=0}^{k}\binom{\frac{n(1-p)}{n+p}}{k-m}\sum_{\begin{subarray}{c}i_{1},\dots,i_{n-1}\geq
0\\\
i_{1}+2i_{2}+\dots+(n-1)i_{n-1}=m\end{subarray}}c(n,p,i(m))\prod\limits_{j=1}^{n-1}{n-1\choose
j}^{i_{j}}.$

We summarize some consequences for the Euclidean ball in the next corollary.

###### Corollary 4.4

$(i)\hskip 11.38109pt\mathcal{V}^{0}_{k}(B^{n}_{2})={n\choose k}{\rm
vol}_{n-1}(\partial B^{n}_{2})=\left\\{\begin{aligned} &{n\choose k}{\rm
vol}_{n-1}(\partial B^{n}_{2})\quad&\text{ if }\ 0\leq k\leq n,\\\
&0\quad&\text{ if }\,k\geq n+1.\end{aligned}\right.$

$(ii)\hskip 11.38109pt\mathcal{V}^{n}_{k}(B^{n}_{2})=C(n,n,k)\,{\rm
vol}_{n-1}(\partial B^{n}_{2})=\left\\{\begin{aligned} &{\rm
vol}_{n-1}(\partial B^{n}_{2})\quad&\text{ if }\ k=0,\\\ &0\quad&\text{ if
}\,k\geq 1.\end{aligned}\right.$

$\hskip-142.26378pt(iii)\hskip 11.38109pt\mathcal{V}_{k}^{p}(B^{n}_{2})={\rm
vol}_{n-1}(\partial B^{n}_{2}){n\frac{n-p}{n+p}\choose k}.$

We now analyze the expressions $C(n,p,k)$ further. This is also needed to
determine continuity issues of $L_{p}$-Steiner quermassintegrals in Section
5.3. We start with the case $p=1$.

###### Proposition 4.5

The following holds for all $n\geq 2$:

  * (i)

$C(n,1,k)>0$ for $k\leq n-1$;

  * (ii)

$(-1)^{k-n+1}C(n,1,k)>0$ for $k\geq n$.

Proof

  * (i)

Denote $\alpha=n\frac{n-1}{n+1}$. Using (40),

$C(n,1,k)=\binom{\alpha}{k}=\frac{\alpha(\alpha-1)(\alpha-2)\cdots(\alpha-k+1)}{k(k-1)\cdots
1}.$ (41)

It is clear that the denominator is positive. We show that the numerator is
also positive when $k\leq n-1$. Note that $\alpha-j=n-2+\frac{2}{n+1}-j$ for
$1\leq j\leq k-1$. Then $\alpha-j>0$ when $j\leq n-2$. Since $k\leq n-1$, $j$
can change in the interval $1\leq j\leq n-2$ which implies that $C(n,1,k)>0.$

  * (ii)

The numerator of (41) consists of $k-1$ factors $\alpha-j$ each of which is
positive when $j\leq n-2$ and negative when $j\geq n-1.$ If $k\geq n$, the
first $n-2$ terms of (41) are positive, and the remaining $k-n+1$ terms are
negative which implies the statement.

Now we address the general case.

###### Proposition 4.6

The following holds for all $n\geq 2$:

  * (i)

If $p<-n$, then $(-1)^{k}C(n,p,k)>0$ for $k\in\mathbb{N}$;

  * (ii)

If $-n<p<n-2+\frac{2}{n+1}$, then $C(n,p,k)>0$ for all $k\leq\lfloor
n-2p+\frac{2p^{2}}{n+p}\rfloor+1$ and $(-1)^{k-(\lfloor
n-2p+\frac{2p^{2}}{n+p}\rfloor+1)}C(n,p,k)~{}>~{}0$ for $k>\lfloor
n-2p+\frac{2p^{2}}{n+p}\rfloor+1$;

  * (iii)

If $n-2+\frac{2}{n+1}\leq p<n$, then $(-1)^{k-1}C(n,p,k)>0$ for
$k\in\mathbb{N}$;

  * (iv)

If $p=n$, then $C(n,p,0)=1$ and $C(n,p,k)=0$ for $k\in\mathbb{N}$;

  * (v)

If $p>n$, then $(-1)^{k}C(n,p,k)>0$ for $k\in\mathbb{N}$.

Proof  Denote $\alpha=n\frac{n-p}{n+p}$. By (40),

$C(n,p,k)=\binom{\alpha}{k}=\frac{\alpha(\alpha-1)(\alpha-2)\cdots(\alpha-k+1)}{k(k-1)\cdots
1}.$ (42)

  * (i)

Since denominator of (42) is positive, numerator of (42) determines the sign
of $C(n,p,k)$. If $p<-n$, then $\alpha<0$ which implies that $\alpha-j<0$ for
all $1\leq j\leq k-1$. As there are $k$ negative terms in the numerator, we
get the result.

  * (ii)

Here $\alpha-j>0$ when $j\leq\lfloor n-2p+\frac{2p^{2}}{n+p}\rfloor$, and
$\alpha-j<0$ otherwise. Since $1\leq j\leq k-1$, if $k\leq\lfloor
n-2p+\frac{2p^{2}}{n+p}\rfloor+1$, then $\alpha-j>0$ which yields that
$C(n,p,k)>0$. If $k>\lfloor n-2p+\frac{2p^{2}}{n+p}\rfloor+1$, then the first
$\lfloor n-2p+\frac{2p^{2}}{n+p}\rfloor$ terms $\alpha-j$ are positive and the
remaining $k-(\lfloor n-2p+\frac{2p^{2}}{n+p}\rfloor+1)$ terms are negative
which implies the result.

  * (iv)

If $p=n$, then $\alpha=0$, $C(n,p,0)=\binom{0}{0}=1$ and
$C(n,p,k)=\binom{0}{k}=0$ for $k\in\mathbb{N}$.

The proof of (iii) and (v) is similar to proof of (i). Remark In the special
case when $p=\frac{n(n-l)}{n+l}$, $l\in\mathbb{N}$, $\frac{n(n-p)}{n+p}$ is an
integer. This simplifies the binomial coefficients
$\binom{\frac{n(n-p)}{n+p}}{k}$ that appear in (23). Thus, the coefficients
$C(n,p,k)>0$ for $k\leq l$ and $C(n,p,k)=0$ for $k>l$.

##  5 Properties of the $L_{p}$-Steiner quermassintegrals

###  5.1 Homogeneity and Invariance

###### Theorem 5.1

Let $K$ be a convex body in $\mathcal{C}^{n}$ and let $p\in\mathbb{R}$ be such
that $p\geq 0$ or $p<-n$. Then for all $k\in\mathbb{N}$, we have

  * (i)

$\mathcal{W}^{p}_{m,\,k}(K)$ and $\mathcal{V}^{p}_{k}(K)$ are homogeneous of
degree $n\frac{n-p}{n+p}-k$.

  * (ii)

$\mathcal{W}^{p}_{m,\,k}(K)$ and $\mathcal{V}^{p}_{k}(K)$ are invariant under
rotations and reflections.

Remark Property (i) of Theorem 5.1 implies that for the Euclidean ball with
radius $r$,

$\mathcal{V}^{p}_{k}(rB^{n}_{2})=r^{n\frac{n-p}{n+p}-k}\,\mathcal{V}^{p}_{k}(B^{n}_{2})=r^{n\frac{n-p}{n+p}-k}\,{\rm
vol}_{n-1}(\partial B^{n}_{2})\,C(n,p,k).$ (43)

The following corollary, for the case $p=1$, immediately follows from Theorem
5.1. In addition to rotation- and reflection-invariance, we also have
invariance under translations.

###### Corollary 5.2

Let $K$ be a convex body in $\mathcal{C}^{n}$. Then for all $k\in\mathbb{N}$,
$\mathcal{V}^{1}_{k}=\mathcal{W}_{k,\,k}$ are invariant under rigid motions
and homogeneous of degree $n\frac{n-1}{n+1}-k$.

####  5.1.1 Proof of Theorem 5.1

To show Theorem 5.1, we need the following facts (see, e.g., [33]).

###### Proposition 5.3

Let $g:\partial K\rightarrow\mathbb{R}$ be an integrable function, and
$T:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ be an invertible, linear map. Then

$\int_{\partial K}g(x)\,d\mathcal{H}^{n-1}(x)=|\det(T)|^{-1}\,\int_{\partial
T(K)}\|T^{-1t}(N_{K}(T^{-1}(y)))\|^{-1}g(T^{-1}(y))\,d\mathcal{H}^{n-1}(x)$

and

$\langle T^{-1}(y,)N_{K}(T^{-1}y)\rangle=\langle
y,N_{T(K)}(y)\rangle\,\|T^{-1t}(N_{K}(T^{-1}(y)))\|$

for all $y\in\partial T(K)$.

We apply Proposition 5.3 to $\mathcal{W}^{p}_{m,\,k}(K)$ and get

$\displaystyle\mathcal{W}^{p}_{m,\,k}(K)=$
$\displaystyle|\det(T)|^{-1}\,\int\limits_{\partial
T(K)}\|T^{-1t}(N_{K}(T^{-1}(y)))\|^{m-k-1+\frac{n(1-p)}{n+p}}\langle
y,N_{T(K)}(y)\rangle^{m-k+\frac{n(1-p)}{n+p}}H_{n-1}^{\frac{p}{n+p}}(T^{-1}(y))$
$\displaystyle\sum_{\begin{subarray}{c}i_{1},\dots,i_{n-1}\geq 0\\\
i_{1}+2i_{2}+\dots+(n-1)i_{n-1}=m\end{subarray}}c(n,p,i(m))\prod\limits_{j=1}^{n-1}{n-1\choose
j}^{i_{j}}H_{j}^{i_{j}}\left(T^{-1}(y)\right)\,d\mathcal{H}^{n-1}(y).$ (44)

(i) Let $a\in\mathbb{R}$. We apply (5.1.1) to $T=a\,Id$. We also use that for
$y\in\partial(aK)$,

$H_{j}(y)=\frac{H_{j}(T^{-1}y)}{a^{j}}.$

We get

$\displaystyle\mathcal{W}^{p}_{m,\,k}(K)=a^{k-n\frac{n-p}{n+p}}\,$
$\displaystyle\int\limits_{\partial aK}\langle
y,N_{aK}(y)\rangle^{m-k+\frac{n(1-p)}{n+p}}H_{n-1}^{\frac{p}{n+p}}(y)\sum_{\begin{subarray}{c}i_{1},\dots,i_{n-1}\geq
0\\\
i_{1}+2i_{2}+\dots+(n-1)i_{n-1}=m\end{subarray}}c(n,p,i(m))\prod\limits_{j=1}^{n-1}{n-1\choose
j}^{i_{j}}H_{j}^{i_{j}}\left(y\right)\,d\mathcal{H}^{n-1}(y)$
$\displaystyle=a^{k-n\frac{n-p}{n+p}}\,\mathcal{W}^{p}_{m,\,k}(aK).$

Consequently,

$\mathcal{V}^{p}_{k}(K)=\sum\limits_{m=0}^{k}\binom{\frac{n(1-p)}{n+p}}{k-m}\mathcal{W}^{p}_{m,\,k}(K)=a^{k-n\frac{n-p}{n+p}}\,\sum\limits_{m=0}^{k}\binom{\frac{n(1-p)}{n+p}}{k-m}\mathcal{W}^{p}_{m,\,k}(aK)=a^{k-n\frac{n-p}{n+p}}\,\mathcal{V}^{p}_{k}(aK).$

(ii) If $T$ is a rotation or a reflection, then $|\det T|=1$,
$\|T^{-1t}(N_{K}(T^{-1}(y)))\|=\|N_{K}(T^{-1}(y))\|=1$ and for all $1\leq
j\leq n-1$,

$\\{H_{j}(y):y\in\partial T(K)\\}=\\{H_{j}\left(x\right):\,x\in\partial K\\}.$

Thus

$\displaystyle\mathcal{W}^{p}_{m,\,k}(K)=$ $\displaystyle\int\limits_{\partial
T(K)}\langle
y,N_{T(K)}(y)\rangle^{m-k+\frac{n(1-p)}{n+p}}H_{n-1}^{\frac{p}{n+p}}(y)\sum_{\begin{subarray}{c}i_{1},\dots,i_{n-1}\geq
0\\\
i_{1}+2i_{2}+\dots+(n-1)i_{n-1}=m\end{subarray}}c(n,p,i(m))\prod\limits_{j=1}^{n-1}{n-1\choose
j}^{i_{j}}H_{j}^{i_{j}}\left(y\right)\,d\mathcal{H}^{n-1}(y)$
$\displaystyle=\mathcal{W}^{p}_{m,\,k}(T(K)),$

which implies

$\mathcal{V}^{p}_{k}(K)=\sum\limits_{m=0}^{k}\binom{\frac{n(1-p)}{n+p}}{k-m}\mathcal{W}^{p}_{m,\,k}(K)=\sum\limits_{m=0}^{k}\binom{\frac{n(1-p)}{n+p}}{k-m}\mathcal{W}^{p}_{m,\,k}(T(K))=\mathcal{V}^{p}_{k}(T(K)).$

Remarks 1\. Under the additional assumption that $K$ is $C^{2}_{+}$, the proof
of the homogeneity property is an immediate consequence of the homogeneity of
$L_{p}$ affine surface area and (15). Indeed, we have

$as_{p}(\lambda
K+tB^{n}_{2})=as_{p}\left(\lambda\left(K+\frac{t}{\lambda}B^{n}_{2}\right)\right)=\lambda^{n\frac{n-p}{n+p}}as_{p}\left(K+\frac{t}{\lambda}B^{n}_{2}\right)=\sum\limits_{k=0}^{\infty}\mathcal{V}^{p}_{k}(K)\lambda^{n\frac{n-p}{n+p}-k}t^{k}.$

On the other hand, using (15) directly, we get

$as_{p}(\lambda K+tB)=\sum\limits_{k=0}^{\infty}\mathcal{V}^{p}_{k}(\lambda
K)t^{k}.$

Therefore,

$\mathcal{V}^{p}_{k}(\lambda
K)=\lambda^{n\frac{n-p}{n+p}-k}\mathcal{V}^{p}_{k}(K).$

2\. We cannot expect that $\mathcal{V}^{p}_{k}(K),\mathcal{W}^{p}_{m,\,k}(K)$
are invariant under general linear transformations. For instance, by (5.1.1),
and also using (see [33]) that

$H_{n-1}(x)=\|T^{-1t}(N_{K}(x))\|^{n+1}|\det(T)|^{2}H_{n-1}(T(x))$

we have

$\displaystyle\mathcal{V}^{p}_{1}(K)$ $\displaystyle=$
$\displaystyle\frac{n(1-p)}{n+p}\mathcal{W}^{p}_{0,\,1}(K)+\mathcal{W}^{p}_{1,\,1}(K)=\frac{n}{n+p}|\det(T)|^{-\frac{n-p}{n+p}}\,$
$\displaystyle\Bigg{(}(1-p)\,\int\limits_{\partial
T(K)}\|T^{-1t}(N_{K}(T^{-1}(y)))\|^{-1}\langle
y,N_{T(K)}(y)\rangle^{-1+\frac{n(1-p)}{n+p}}H_{n-1}^{\frac{p}{n+p}}(y)\,d\mathcal{H}^{n-1}(y)$
$\displaystyle+$ $\displaystyle(n-1)\int\limits_{\partial T(K)}\langle
y,N_{T(K)}(y)\rangle^{\frac{n(1-p)}{n+p}}H_{n-1}^{\frac{p}{n+p}}(y)H_{1}(T^{-1}(y))\,d\mathcal{H}^{n-1}(y)\Bigg{)}$

In particular, if $n=2$, then $H_{n-1}=H_{1}$ and thus

$\displaystyle\mathcal{V}^{p}_{1}(K)$ $\displaystyle=$
$\displaystyle\frac{2(1-p)}{2+p}\mathcal{W}^{p}_{0,\,1}(K)+\mathcal{W}^{p}_{1,\,1}(K)=$
$\displaystyle\hskip-31.29802pt\frac{2}{2+p}|\det(T)|^{-\frac{2-p}{2+p}}\Bigg{(}(1-p)\,\int\limits_{\partial
T(K)}\|T^{-1t}(N_{K}(T^{-1}(y)))\|^{-1}\langle
y,N_{T(K)}(y)\rangle^{-1+\frac{2(1-p)}{2+p}}H_{1}^{\frac{2}{2+p}}(y)\,d\mathcal{H}^{1}(y)$
$\displaystyle\hskip-31.29802pt+|\det(T)|^{2}\int\limits_{\partial
T(K)}\|T^{-1t}(N_{K}(T^{-1}(y)))\|^{3}\langle
y,N_{T(K)}(y)\rangle^{\frac{2(1-p)}{2+p}}H_{1}^{\frac{2(p+1)}{2+p}}(y)\,d\mathcal{H}^{1}(y)\Bigg{)},$

while

$\displaystyle\mathcal{V}^{p}_{1}(T(K))=\frac{2(1-p)}{2+p}\mathcal{W}^{p}_{0,\,1}(T(K))+\mathcal{W}^{p}_{1,\,1}(T(K))=$
$\displaystyle\hskip-28.45274pt\frac{2}{2+p}\Bigg{(}(1-p)\int\limits_{\partial
T(K)}\langle
y,N(y)\rangle^{-1+\frac{2(1-p)}{2+p}}H_{1}^{\frac{2}{2+p}}(y)\,d\mathcal{H}^{1}(y)+\int\limits_{\partial
T(K)}\langle
y,N(y)\rangle^{\frac{2(1-p)}{2+p}}H_{1}^{\frac{2(p+1)}{2+p}}(y)\,d\mathcal{H}^{1}(y)\Bigg{)}.$

That is, unless $T$ is an isometry, we cannot expect to have invariance.

###  5.2 Valuation property

For $p=1$, we have the following

###### Theorem 5.4

For all $k\in\mathbb{N}$, $\mathcal{W}_{k,\,k}$ are valuations on the set
$\mathcal{C}^{n}$.

More generally, we obtain

###### Theorem 5.5

Let $p\in\mathbb{R}$ be such that $p\geq 0$ or $p<-n$. For all
$k,m\in\mathbb{N}$, $\mathcal{W}_{m,\,k}^{p}$ and $\mathcal{V}^{p}_{k}$ are
valuations on the set $\mathcal{C}^{n}$.

####  5.2.1 Proof of Theorem 5.4 and Theorem 5.5

We need the following lemma which can be found in e.g. [31].

###### Lemma 5.6

Let $C$ and $K$ be convex bodies in $\mathbb{R}^{n}$ and suppose that $C\cup
K$ is a convex body. Then we have for all $x\in\partial C\cap\partial K$ and
for all $\alpha\geq 0$ where the principal curvatures $k_{j}(\partial(C\cup
K),x)$, $k_{j}(\partial(C\cap K),x)$, $k_{j}(C,x)$ and $k_{j}(K,x)$ exist for
all $1\leq j\leq n-1$,

$\displaystyle H_{n-1}(\partial(C\cup
K),x)^{\frac{1}{n+1}}k_{j}(\partial(K\cup
C),x)^{\alpha}=\min\\{H_{n-1}(C,x)^{\frac{1}{n+1}}k_{j}(C,x)^{\alpha},H_{n-1}(K,x)^{\frac{1}{n+1}}k_{j}(K,x)^{\alpha}\\}$

and

$\displaystyle H_{n-1}(\partial(C\cap
K),x)^{\frac{1}{n+1}}k_{j}(\partial(K\cap
C),x)^{\alpha}=\max\\{H_{n-1}(C,x)^{\frac{1}{n+1}}k_{j}(C,x)^{\alpha},H_{n-1}(K,x)^{\frac{1}{n+1}}k_{j}(K,x)^{\alpha}\\}.$

Moreover, for all $1\leq i_{1},\dots,i_{j}\leq n-1$, and for all
$\alpha_{1},\alpha_{2},\dots,\alpha_{j}\geq 0$,

$\displaystyle H_{n-1}(\partial(C\cup
K),x)^{\frac{1}{n+1}}\prod_{i=1}^{j}k_{i_{j}}(\partial(K\cup
C),x)^{\alpha_{j}}=$
$\displaystyle\min\\{H_{n-1}(C,x)^{\frac{1}{n+1}}\prod_{i=1}^{j}k_{i_{j}}(C,x)^{\alpha_{j}},H_{n-1}(K,x)^{\frac{1}{n+1}}\prod_{i=1}^{j}k_{i_{j}}(K,x)^{\alpha_{j}}\\}$

and

$\displaystyle H_{n-1}(\partial(C\cap
K),x)^{\frac{1}{n+1}}\prod_{i=1}^{j}k_{i_{j}}(\partial(K\cap
C),x)^{\alpha_{j}}=$
$\displaystyle\max\\{H_{n-1}(C,x)^{\frac{1}{n+1}}\prod_{i=1}^{j}k_{i_{j}}(C,x)^{\alpha_{j}},H_{n-1}(K,x)^{\frac{1}{n+1}}\prod_{i=1}^{j}k_{i_{j}}(K,x)^{\alpha_{j}}\\}.$

###### Theorem 5.7

Let $p\in\mathbb{R}$ be such that $p\geq 0$ or $p<-n$. For all $1\leq
i_{1},\dots,i_{j}\leq n-1$ and
$\alpha_{1},\alpha_{2},\dots,\alpha_{j}~{}\geq~{}0$,

$\int\limits_{\partial K}H_{n-1}^{\frac{1}{n+1}}(x)\
\prod_{i=1}^{j}k_{i_{j}}^{\alpha_{j}}(x)\,d\mathcal{H}^{n-1}(x)$

are valuations on the set $\mathcal{C}^{n}$.

Proof  Since $C$ and $K$ are in $\mathcal{C}^{n}$, then $C\cap K$ and $C\cup
K$ are in $\mathcal{C}^{n}$ by Proposition 4.2. To prove the valuation
property, we follow the approach of [31]. Let $C$ and $K$ be convex bodies in
$\mathbb{R}^{n}$ such that $C\cup K$ is a convex body. As above, we decompose

$\displaystyle\partial(C\cup K)$ $\displaystyle=$ $\displaystyle\\{\partial
C\cap\partial K\\}\cup\\{\partial C\cap K^{c}\\}\cup\\{C^{c}\cap\partial K\\}$
$\displaystyle\partial(C\cap K)$ $\displaystyle=$ $\displaystyle\\{\partial
C\cap\partial K\\}\cup\\{\partial
C\cap\operatorname{int}(K)\\}\cup\\{\operatorname{int}(C)\cap\partial K\\}$
$\displaystyle\partial C$ $\displaystyle=$ $\displaystyle\\{\partial
C\cap\partial K\\}\cup\\{\partial C\cap K^{c}\\}\cup\\{\partial
C\cap\operatorname{int}(K)\\}$ $\displaystyle\partial K$ $\displaystyle=$
$\displaystyle\\{\partial C\cap\partial K\\}\cup\\{\partial K\cap
C^{c}\\}\cup\\{\partial K\cap\operatorname{int}(C)\\}.$

Since all the sets (except possibly $\partial C\cap\partial K$) in the right-
hand side of the above decomposition are open subsets of $\partial C$,
$\partial K$, $\partial(C\cap K)$ and $\partial(C\cup K)$, the integrals over
those sets cancel out. Below we use $d\mathcal{H}^{n-1}_{K}(x)$ to denote the
usual surface area measure on $\partial K$ and $k_{j}(K,x)$ to emphasize that
this is the principal curvature of $K$ at point $x$. It remains to show

$\displaystyle\int_{\partial C\cap\partial K}H_{n-1}(\partial(C\cup
K),x)^{\frac{1}{n+1}}\prod_{i=1}^{j}k_{i_{j}}(\partial(K\cup
C),x)^{\alpha_{j}}d\mathcal{H}^{n-1}_{C\cup K}(x)$
$\displaystyle+\int_{\partial C\cap\partial K}H_{n-1}(\partial(C\cap
K),x)^{\frac{1}{n+1}}\prod_{i=1}^{j}k_{i_{j}}(\partial(K\cap
C),x)^{\alpha_{j}}d\mathcal{H}^{n-1}_{C\cap K}(x)$
$\displaystyle=\int_{\partial C\cap\partial
K}H_{n-1}(C,x)^{\frac{1}{n+1}}\prod_{i=1}^{j}k_{i_{j}}(C,x)^{\alpha_{j}}d\mathcal{H}^{n-1}_{C}(x)+\int_{\partial
C\cap\partial
K}H_{n-1}(K,x)^{\frac{1}{n+1}}\prod_{i=1}^{j}k_{i_{j}}(K,x)^{\alpha_{j}}d\mathcal{H}^{n-1}_{K}(x).$

Please note that $d\mathcal{H}^{n-1}_{C\cup K}=d\mathcal{H}^{n-1}_{C\cap K}$
on $\partial K\cap\partial C$. This holds because both measure are equal to
the $(n-1)$-dimensional Hausdorff measure. Therefore, it is left to show

$\displaystyle\int_{\partial C\cap\partial K}H_{n-1}(\partial(C\cup
K),x)^{\frac{1}{n+1}}\prod_{i=1}^{j}k_{i_{j}}(\partial(K\cup
C),x)^{\alpha_{j}}d\mathcal{H}^{n-1}_{C\cap K}(x)$
$\displaystyle+\int_{\partial C\cap\partial K}H_{n-1}(\partial(C\cap
K),x)^{\frac{1}{n+1}}\prod_{i=1}^{j}k_{i_{j}}(\partial(K\cap
C),x)^{\alpha_{j}}d\mathcal{H}^{n-1}_{C\cap K}(x)$
$\displaystyle=\int_{\partial C\cap\partial
K}H_{n-1}(C,x)^{\frac{1}{n+1}}\prod_{i=1}^{j}k_{i_{j}}(C,x)^{\alpha_{j}}d\mathcal{H}^{n-1}_{C}(x)+\int_{\partial
C\cap\partial
K}H_{n-1}(K,x)^{\frac{1}{n+1}}\prod_{i=1}^{j}k_{i_{j}}(K,x)^{\alpha_{j}}d\mathcal{H}^{n-1}_{K}(x).$

By Lemma 5.6 this is equivalent to

$\displaystyle\int_{\partial C\cap\partial
K}\min\\{H_{n-1}(C,x)^{\frac{1}{n+1}}\prod_{i=1}^{j}k_{i_{j}}(C,x)^{\alpha_{j}},H_{n-1}(K,x)^{\frac{1}{n+1}}\prod_{i=1}^{j}k_{i_{j}}(K,x)^{\alpha_{j}}\\}d\mathcal{H}^{n-1}_{C\cap
K}(x)$ $\displaystyle+\int_{\partial C\cap\partial
K}\max\\{H_{n-1}(C,x)^{\frac{1}{n+1}}\prod_{i=1}^{j}k_{i_{j}}(C,x)^{\alpha_{j}},H_{n-1}(K,x)^{\frac{1}{n+1}}\prod_{i=1}^{j}k_{i_{j}}(K,x)^{\alpha_{j}}\\}d\mathcal{H}^{n-1}_{C\cap
K}(x)$ $\displaystyle=\int_{\partial C\cap\partial
K}H_{n-1}(C,x)^{\frac{1}{n+1}}\prod_{i=1}^{j}k_{i_{j}}(C,x)^{\alpha_{j}}d\mathcal{H}^{n-1}_{C}(x)+\int_{\partial
C\cap\partial
K}H_{n-1}(K,x)^{\frac{1}{n+1}}\prod_{i=1}^{j}k_{i_{j}}(K,x)^{\alpha_{j}}d\mathcal{H}^{n-1}_{K}(x).$

This holds since for any real numbers $a$ and $b$, we have

$a+b=\min\\{a,b\\}+\max\\{a,b\\}.$

We note that in the setting of Lemma 5.6, $N_{K}(x)=N_{C}(x)=N_{C\cap
K}(x)=N_{C\cup K}(x)$. Using this observation together with Lemma 5.6 and the
decomposition in the proof of Theorem 5.7, we get the following
generalization.

###### Theorem 5.8

Let $p\in\mathbb{R}$ be such that $p\geq 0$ or $p<-n$. For all $1\leq
i_{1},\dots,i_{j}\leq n-1$ and
$\alpha_{1},\alpha_{2},\dots,\alpha_{j}~{}\geq~{}0$,

$\int\limits_{\partial K}\langle
x,N(x)\rangle^{m-k+\frac{n(1-p)}{n+p}}H_{n-1}^{\frac{p}{n+p}}(x)\
\prod_{i=1}^{j}k_{i_{j}}^{\alpha_{j}}(x)\,d\mathcal{H}^{n-1}(x)$

are valuations on the set $\mathcal{C}^{n}$.

Proof of Theorem 5.4 The result follows immediately from Theorem 5.7 as

$\mathcal{W}_{k,\,k}(K)=\int\limits_{\partial
K}H_{n-1}^{\frac{1}{n+1}}(x)\sum_{\begin{subarray}{c}i_{1},\dots,i_{n-1}\geq
0\\\
i_{1}+2i_{2}+\dots+(n-1)i_{n-1}=k\end{subarray}}c(n,1,i(k))\prod\limits_{j=1}^{n-1}{n-1\choose
j}^{i_{j}}H_{j}^{i_{j}}\left(x\right)\,d\mathcal{H}^{n-1}(x)$

is a sum (up to constants) of integrals of the form

$\int\limits_{\partial K}H_{n-1}^{\frac{1}{n+1}}(x)\
\prod_{i=1}^{j}k_{i_{j}}^{\alpha_{j}}(x)\,d\mathcal{H}^{n-1}(x)$

and as the sum of valuations, it is again a valuation.

Proof of Theorem 5.5 The result follows immediately from Theorem 5.8 and the
fact that the linear combination of valuations is again a valuation.

Remark When $j=n-1$, Theorem 5.7 states that for any
$\alpha_{1},\dots,\alpha_{n-1}\geq 0$

$\int\limits_{\partial
K}H^{\frac{1}{n+1}}_{n-1}(x)\prod_{i=1}^{n-1}k_{i}^{\alpha_{i}}(x)\,d\mathcal{H}^{n-1}(x)$

is a valuation on the set $\mathcal{C}^{n}$. In particular, when
$\alpha_{1}=\dots=\alpha_{n-1}=\frac{s}{n+1}$ for any $s>0$, then

$\int\limits_{\partial
K}H^{\frac{1}{n+1}}_{n-1}(x)\left(\prod_{i=1}^{n-1}k_{i}(x)\right)^{\frac{s}{n+1}}\,d\mathcal{H}^{n-1}(x)=\int\limits_{\partial
K}H^{\frac{s+1}{n+1}}_{n-1}(x)\,d\mathcal{H}^{n-1}(x)=as_{1,s}(K)$

and Theorem 5.7 implies that mixed affine surface areas $as_{1,s}$ are
valuations. It seems that this fact was not known before.

###  5.3 Continuity

For convex bodies $K$ and $L$, their Hausdorff distance is

$d_{H}(K,L)=\min\\{\varepsilon:K\subset L+\varepsilon B_{2}^{n},L\subset
K+\varepsilon B_{2}^{n}\\}.$ (45)

It was proved by Lutwak [20] (see also [14]) that for $p\geq 1$, $L_{p}$
affine surface area is an upper semi continuous functional with respect to the
Hausdorff metric. In fact, it follows from Lutwak’s proof that the same holds
for all $0\leq p<1$ (aside from the case $p=0$, which is just volume and hence
continuous). For $-n<p\leq 0$, the functional is lower semi continuous as was
shown by Ludwig [15]. It was also shown there that the functional defined by
(11) is not lower semi continuous when $-n<p\leq 0$, and that it is lower semi
continuous for $p<-n$ whereas such a statement is not true in this range for
the functional defined by (8).

As $\mathcal{V}_{0}^{p}(K)=\mathcal{W}^{p}_{0,\,0}=as_{p}(K)$, it is natural
to ask about the continuity properties for the $L_{p}$-Steiner
quermassintegrals $\mathcal{V}_{k}^{p}$ and the $L_{p}$ Steiner coefficients
$\mathcal{W}^{p}_{m,\,k}$. Of course, if $p=0$, then
$\mathcal{V}_{0}^{0}(K)=n\,{\rm vol}_{n}(K)$, which is continuous. We first
consider the cases when the assumption $K\in\mathcal{C}^{n}$ is not needed.
This holds for $\mathcal{W}^{p}_{0,\,k}$ (30). As these coincide with the
mixed $L_{p}$ affine surface areas $as_{p+\frac{k}{n}(n+p),-k}$, this shows
the continuity properties of the latter, which, as far as we know, had not
been stated before.

###### Theorem 5.9

Let $k\geq 0$.

  * (i)

For $0\leq p\leq\infty$, $\mathcal{W}^{p}_{0,\,k}$ is upper semi continuous on
the set $\mathcal{K}^{n}_{o}$ and for $-n<p\leq 0$, lower semi continuous on
the set $\mathcal{K}^{n}_{o}$.

  * (ii)

The $as_{p,s}(K)$ are upper semi continuous on the set $\mathcal{K}^{n}_{o}$
when $s<0$ and $p>-s$, and lower semi continuous on the set
$\mathcal{K}^{n}_{o}$ if $s<0$ and $-n<p<-s$.

Proof  We prove part (i). Part (ii) follows similarly. We note that if $K_{l}$
converges to $K$ in the Hausdorff metric, then $h_{K_{l}}$ converges to
$h_{K}$ uniformly on $S^{n-1}$. Using 30, the proof of (i) then follows from
the proof of upper semi continuity of a generalized $L_{p}$ affine surface
area given in [14] where we take $f(t)=t^{\frac{p}{n+p}}$ that satisfies the
conditions in [14] (since $0<\frac{p}{n+p}<1$). The next corollary follows
immediately from this theorem and the previous sections. A characterization of
upper semi continuous rigid motion invariant valuations on convex bodies in
the plane was given in [13].

###### Corollary 5.10

Let $p\geq 0$. Then $\mathcal{W}^{p}_{0,\,k}=as_{p+\frac{k}{n}(n+p),-k}$ are
upper semi continuous $n\frac{n-p}{n+p}~{}-~{}k$ homogeneous valuations on the
set $\mathcal{K}^{n}_{o}$ that are invariant under rotations and reflections.

We will repeatedly use the following example:
For $x\in\mathbb{R}^{n}$, let $\|x\|_{\infty}=\max_{1\leq i\leq n}|x_{i}|$ and
let $B^{n}_{\infty}=\\{x\in\mathbb{R}^{n}:\|x\|_{\infty}\leq 1\\}$. For
$l\in\mathbb{N}$, we consider convex bodies $K_{l}$. We describe $K_{l}$ for
the first quadrant $\mathbb{R}^{n}_{+}$, and call them $K_{l}^{+}$. The other
quadrants are described accordingly. Let
$x_{0}=(1-\frac{1}{l},\cdots,1-\frac{1}{l})$ and
$B^{n}_{2}\left(x_{0},\frac{1}{l}\right)$ be the Euclidean ball centered at
$x_{0}$ with radius $\frac{1}{l}$. We put

$K_{l}^{+}=\left(1-\frac{1}{l}\right)B^{n}_{\infty}+\frac{1}{l}B^{n}_{2}.$
(46)

Remark Let $K_{l}$ be as in (46). Then $K_{l}\rightarrow B^{n}_{\infty}$ in
the Hausdorff metric and $\mathcal{W}^{p}_{0,\,k}(B^{n}_{\infty})=0$ for
$-\infty\leq p<-n$. Moreover,

$\displaystyle\mathcal{W}^{p}_{0,\,k}(K_{l})$ $\displaystyle=$
$\displaystyle\int\limits_{\partial K_{l}}H_{n-1}^{\frac{p}{n+p}}\langle
x,N(x)\rangle^{-k-n\frac{p-1}{n+p}}d\mathcal{H}^{n-1}(x)$ $\displaystyle\geq$
$\displaystyle
n^{\frac{1}{2}\left(-k-n\frac{p-1}{n+p}\right)}\,l^{-n\frac{n-p}{n+p}}{\rm
vol}_{n-1}\,\,(\partial B^{n}_{2})\rightarrow\infty,$

as $l\rightarrow\infty$. This shows that $\mathcal{W}^{p}_{0,\,k}$ is not
upper semi continuous for $-\infty\leq p<-n$. It is also not lower semi
continuous for that $p$-range as is easily seen by taking a sequence of
polytopes that converges to the Euclidean unit ball.

While there are continuity properties of the $L_{p}$ Steiner coefficients for
$m=0$, we cannot expect that any of the other $L_{p}$ Steiner coefficients or
$L_{p}$-Steiner quermassintegrals have continuity properties.

We note that $\mathcal{C}^{n}$ is not closed under the topology generated by
the Hausdorff metric. Thus, when addressing continuity issues, we consider
convergence in the Hausdorff topology involving bodies for which the $L_{p}$
Steiner coefficients are well-defined but such that these bodies are not
necessarily in $\mathcal{C}^{n}$.

###### Proposition 5.11

Let $k\geq 1$. Then $\mathcal{V}^{1}_{k}=\mathcal{W}_{k,\,k}$ are neither
lower semi continuous nor upper semi continuous.

Proof  It is clear that $\mathcal{W}_{k,\,k}$ is not continuous with respect
to the Hausdorff metric: take a sequence of polytopes that approximates the
Euclidean ball.

Let $K_{l}$ be as in (46). Then $K_{l}\rightarrow B^{n}_{\infty}$ in the
Hausdorff metric and $\mathcal{W}_{k,\,k}(B^{n}_{\infty})=0$. Moreover, even
though $K_{l}$ is not in $\mathcal{C}_{p}$, the coefficients
$\mathcal{W}_{k,\,k}(K_{l})$ are well-defined. Indeed, using (35),

$\displaystyle\mathcal{W}_{k,\,k}(K_{l})$ $\displaystyle=$
$\displaystyle\int\limits_{\partial
K_{l}}H_{n-1}^{\frac{1}{n+1}}\sum_{\begin{subarray}{c}i_{1},\dots,i_{n-1}\geq
0\\\
i_{1}+2i_{2}+\dots+(n-1)i_{n-1}=k\end{subarray}}c(n,1,i(k))\prod\limits_{j=1}^{n-1}{n-1\choose
j}^{i_{j}}H_{j}^{i_{j}}\left(x\right)\,d\mathcal{H}^{n-1}(x)$ (47)
$\displaystyle=$ $\displaystyle\int\limits_{\partial
B^{n}_{2}\left(0,\frac{1}{l}\right)}H_{n-1}^{\frac{1}{n+1}}\sum_{\begin{subarray}{c}i_{1},\dots,i_{n-1}\geq
0\\\
i_{1}+2i_{2}+\dots+(n-1)i_{n-1}=k\end{subarray}}c(n,1,i(k))\prod\limits_{j=1}^{n-1}{n-1\choose
j}^{i_{j}}H_{j}^{i_{j}}\left(x\right)\,d\mathcal{H}^{n-1}(x)$ $\displaystyle=$
$\displaystyle\mathcal{W}_{k,\,k}\left(B^{n}_{2}\left(0,\frac{1}{l}\right)\right)$
$\displaystyle=$ $\displaystyle l^{k-n\frac{n-1}{n+1}}{\rm
vol}_{n-1}\,\,(\partial B^{n}_{2})\,C(n,1,k).$

Let first $k\geq n$. If $k-n+1$ is even, then we have by Proposition 4.5 that
$C(n,1,k)>0$. Therefore, as $k-n\frac{n-1}{n+1}>0$,

$\mathcal{W}_{k,\,k}(K_{l})=l^{k-n\frac{n-1}{n+1}}{\rm vol}_{n-1}\,\,(\partial
B^{n}_{2})\,C(n,1,k)\to\infty,$

as $l\to\infty$, but $\mathcal{W}_{k,\,k}(B^{n}_{\infty})=0$. Thus
$\mathcal{W}_{k,\,k}$ is not upper semi continuous in this case.
$\mathcal{W}_{k,\,k}$ is also not lower semi continuous. We take a sequence
$P_{l}$ of polytopes that converges to $B^{n}_{2}$ in the Hausdorff metric.
Then $\mathcal{W}_{k,\,k}(P_{l})=0$ for all $l$ but
$\mathcal{W}_{k,\,k}(B^{n}_{2})>0$.

Let now $k\geq n$ be such that $k-n+1$ is odd. Then $C(n,1,k)<0$ by
Proposition 4.5 and thus by (31), $\mathcal{W}_{k,\,k}(B^{n}_{2})={\rm
vol}_{n-1}(\partial B^{n}_{2})\,C(n,1,k)<0$. As $\mathcal{W}_{k,\,k}(P_{l})=0$
for all $l$, where $P_{l}$ is a sequence of polytopes that converges to
$B^{n}_{2}$ in the Hausdorff metric, this shows that $\mathcal{W}_{k,\,k}$ is
not lower semi continuous in this case. $\mathcal{W}_{k,\,k}$ is also not
upper semi continuous. Indeed, $K_{l}\to B^{n}_{\infty}$ as $l\to\infty$ in
the Hausdorff metric and $\mathcal{W}_{k,\,k}(B^{n}_{\infty})=0$. But

$\mathcal{W}_{k,\,k}(K_{l})=l^{k-n\frac{n-1}{n+1}}{\rm vol}_{n-1}\,\,(\partial
B^{n}_{2})\,C(n,1,k)\to-\infty.$

This settles negatively the semi continuity of $\mathcal{W}_{k,\,k}$ for all
$n\geq 2$ and all $k\geq n$. Let now $k\leq n-1$. By (31), we have that
$\mathcal{W}_{k,\,k}(B^{n}_{2})={\rm vol}_{n-1}(\partial
B^{n}_{2})\,C(n,1,k)>0$. As $\mathcal{W}_{k,\,k}(P)=0$ for every polytope $P$,
and as by Proposition 4.5, $C(n,1,k)>0$ for all $n\geq 2$, this shows that
$\mathcal{W}_{k,\,k}$ is not lower semi continuous in this $k$-range.
Moreover, by Proposition 4.5, for every $n\geq 2$, $C(n,1,n-1)>0$. Therefore,
for all $n\geq 2$

$\mathcal{W}_{n-1,\,n-1}(K_{l})=l^{\frac{n-1}{n+1}}\,{\rm
vol}_{n-1}\,\,(\partial B^{n}_{2})\,C(n,1,n-1)\rightarrow\infty,$

as $l\rightarrow\infty$ and thus $\mathcal{W}_{n-1,\,n-1}$ is also not upper
semi continuous, which settles negatively the semi continuity of
$\mathcal{W}_{k,\,k}$ for all $n\geq 2$ and all $k\geq n-1$. In particular,
the case $n=2$ is settled for all $k$. To settle upper semi continuity for all
dimensions and all $k<n-1$ requires further examples.

For instance, for $n=3$, only upper semi continuity for $k=1$ is not yet
settled. To resolve this, we consider

$\mathcal{W}_{1,\,1}(K)=\frac{3}{2}\int_{\partial K}H_{2}(x)^{\frac{1}{4}}\
H_{1}(x)\ d\mathcal{H}^{2}(x).$

We place the body $K_{l}$ in the $xz$-plane and we let
$Z_{l}\subset\mathbb{R}^{3}$ be the body of revolution obtained by rotating
$K_{l}$ about the $z$-axis. Let $x\in\partial Z_{l}$. Then the maximal
principle curvature $\kappa_{1}(x)=l$ and the minimal principle curvature
$\kappa_{2}(x)\geq 1$. Therefore,

$H_{2}(x)^{\frac{1}{4}}\geq l^{\frac{1}{4}}\hskip 8.53581pt\text{and}\hskip
8.53581ptH_{1}(x)\geq\frac{1}{2}\left(l+1\right)$

and consequently

$\mathcal{W}_{1,\,1}(Z_{l})\geq\frac{3}{4}l^{\frac{1}{4}}\left(l+1\right)\int_{\partial
Z_{l}}d\mathcal{H}^{2}(x)\geq 3\pi
l^{\frac{1}{4}}\left(l+1\right)\frac{1-\frac{1}{l}}{l}.$

Thus $\mathcal{W}_{1,\,1}(Z_{l})\to\infty$, as $l\to\infty$, but $Z_{l}$
converges in the Hausdorff metric to the cylinder $Z$ of height $2$ and a
$2$-dimensional Euclidean unit ball as base and $\mathcal{W}_{1,\,1}(Z)=0$.
Thus $\mathcal{W}_{1,\,1}$ is not upper semi continuous and this settles $n=3$
for all $k$.

Modifications of these examples show that $\mathcal{W}_{k,\,k}$ is not upper
semi continuous for all $n\geq 4$ and all $1\leq k<n-1$.

Now we treat general parameters $p$ such that $\frac{p}{n+p}\geq 0$.
$\mathcal{V}_{0}^{0}(K)=n\,{\rm vol}_{n}(K)$ and
$\mathcal{V}_{1}^{0}(K)=c(n){\rm vol}_{n-1}(\partial K)$, where $c(n)$ is a
constant depending only on $n$. Thus, $\mathcal{V}_{0}^{0}$ and
$\mathcal{V}_{1}^{0}$ are continuous. Moreover,
$\mathcal{V}_{0}^{p}(K)=as_{p}(K)$ which, as noted above, is upper semi
continuous when $p\geq 0$ and lower semi continuous for $-n<p\leq 0$. In
general however, we do not have any continuity property.

###### Proposition 5.12

Let $-\infty<p<-n$ or $0\leq p<\infty$, and let $k\geq 1$. Let $K$ be such
that $\mathcal{V}_{k}^{p}$ are well-defined. Then $\mathcal{V}_{k}^{p}$ are in
general neither lower semi continuous nor upper semi continuous.

Proof  If $p=0$ and $k\geq 2$, then by (36), $\mathcal{V}_{k}^{0}(P)=0$ for
every polytope $P$ and by Corollary 4.4,
$\mathcal{V}^{0}_{k}(B^{n}_{2})={n\choose k}{\rm vol}_{n-1}(\partial
B^{n}_{2})$ for $2\leq k\leq n$. Thus, taking a sequence $P_{l}$ of polytopes
that converges to $B^{n}_{2}$, this shows that $\mathcal{V}^{0}_{k}$ is not
lower semi continuous for $2\leq k\leq n$.
If $p\neq 0$, then by (37), $\mathcal{V}_{k}^{p}(P)=0$ for every polytope $P$
and if $p<-n$ or $p>n$ then by (33) and Proposition 4.6,

$\mathcal{V}^{p}_{k}(B^{n}_{2})=C(n,p,k)\,{\rm vol}_{n-1}(\partial
B^{n}_{2})\left\\{\begin{aligned} &<0\quad&\text{ if }\,k\,&\text{ is odd }\\\
&>0\quad&\text{ if }\,k\,&\text{ is even }.\end{aligned}\right.$

Taking a sequence $P_{l}$ of polytopes that converges to $B^{n}_{2}$, shows
that for $-\infty\leq p<-n$ or $n<p\leq\infty$, $\mathcal{V}^{p}_{k}$ is not
upper semi continuous when $k$ is odd and not lower semi continuous when $k$
is even. Similarly, absence of upper resp. lower semi continuity can be
determined for the other $p$-ranges, using Proposition 4.6.

For the semi continuity issues not yet settled, we will now use the convex
bodies $K_{l}$, $l\in\mathbb{N}$ given by (46). To determine
$\mathcal{V}^{p}_{k}(K_{l})$, it is enough in this case to consider
$B_{l}=\partial K_{l}\cap\mathbb{R}^{n}_{+}\cap\partial
B^{n}_{2}(x_{0},\frac{1}{l})$, where
$x_{0}=\left(1-\frac{1}{l}\right)(1,1,\cdots,1)$. Note that for
$x=x_{0}+\frac{1}{l}\xi\in B_{l}$, $\xi\in S^{n-1}_{+}$, we have

$\langle
x,N(x)\rangle=h_{K_{l}}(\xi)=\left(1-\frac{1}{l}\right)\sum_{i=1}^{n}\xi_{i}+\frac{1}{l}.$
(48)

Then

$\displaystyle\mathcal{V}_{k}^{p}(K_{l})$ $\displaystyle=$ $\displaystyle
2^{n}\sum\limits_{m=0}^{k}\binom{\frac{n(1-p)}{n+p}}{k-m}\int\limits_{S^{n-1}_{+}}h_{K_{l}}^{m-k+\frac{n(1-p)}{n+p}}\,\frac{l^{(n-1)\frac{p}{n+p}+m}}{l^{n-1}}$
$\displaystyle\sum_{\begin{subarray}{c}i_{1},\dots,i_{n-1}\geq 0\\\
i_{1}+2i_{2}+\dots+(n-1)i_{n-1}=m\end{subarray}}c(n,p,i(m))\prod\limits_{j=1}^{n-1}{n-1\choose
j}^{i_{j}}\,d\mathcal{H}^{n-1}$ $\displaystyle=$ $\displaystyle
2^{n}\sum\limits_{m=0}^{k}l^{-\frac{n(n-1)}{n+p}+m}\binom{\frac{n(1-p)}{n+p}}{k-m}\sum_{\begin{subarray}{c}i_{1},\dots,i_{n-1}\geq
0\\\
i_{1}+2i_{2}+\dots+(n-1)i_{n-1}=m\end{subarray}}c(n,p,i(m))\prod\limits_{j=1}^{n-1}{n-1\choose
j}^{i_{j}}\,$
$\displaystyle\int\limits_{S^{n-1}_{+}}\left(\left(1-\frac{1}{l}\right)\sum_{i=1}^{n}\xi_{i}+\frac{1}{l}\right)^{m-k+\frac{n(1-p)}{n+p}}\,d\mathcal{H}^{n-1},$

where we used (3) and (48) in the first equality. This shows that
$\mathcal{V}_{k}^{p}(K_{l})$ are finite. Therefore,

$\displaystyle\lim_{l\to\infty}\mathcal{V}_{k}^{p}(K_{l})$ $\displaystyle=$
$\displaystyle
2^{n}\lim_{l\to\infty}\sum\limits_{m=0}^{k}l^{-\frac{n(n-1)}{n+p}+m}\binom{\frac{n(1-p)}{n+p}}{k-m}\sum_{\begin{subarray}{c}i_{1},\dots,i_{n-1}\geq
0\\\
i_{1}+2i_{2}+\dots+(n-1)i_{n-1}=m\end{subarray}}c(n,p,i(m))\prod\limits_{j=1}^{n-1}{n-1\choose
j}^{i_{j}}$
$\displaystyle\lim_{l\to\infty}\int\limits_{S^{n-1}_{+}}\left(\left(1-\frac{1}{l}\right)\sum_{i=1}^{n}\xi_{i}+\frac{1}{l}\right)^{m-k+\frac{n(1-p)}{n+p}}\,d\mathcal{H}^{n-1}.$

As for $\xi\in S^{n-1}_{+}$, $1\leq\sum_{i=1}^{n}\xi_{i}\leq\sqrt{n}$, the
functions under the integral are uniformly in $l$ bounded by an integrable
function and by Lebegue’s Dominated Convergence theorem we can interchange
integration and limit. Therefore

$\displaystyle\lim_{l\to\infty}\int\limits_{S^{n-1}_{+}}\left(\left(1-\frac{1}{l}\right)\sum_{i=1}^{n}\xi_{i}+\frac{1}{l}\right)^{m-k+\frac{n(1-p)}{n+p}}\,d\mathcal{H}^{n-1}=$
$\displaystyle\int\limits_{S^{n-1}_{+}}\lim_{l\to\infty}\left(\left(1-\frac{1}{l}\right)\sum_{i=1}^{n}\xi_{i}+\frac{1}{l}\right)^{m-k+\frac{n(1-p)}{n+p}}\,d\mathcal{H}^{n-1}=\int\limits_{S^{n-1}_{+}}\left(\sum_{i=1}^{n}\xi_{i}\right)^{m-k+\frac{n(1-p)}{n+p}}\,d\mathcal{H}^{n-1}.$

Hence

$\displaystyle\lim_{l\to\infty}\mathcal{V}_{k}^{p}(K_{l})$ $\displaystyle=$
$\displaystyle
2^{n}\lim_{l\to\infty}\sum\limits_{m=0}^{k}l^{-\frac{n(n-1)}{n+p}+m}\binom{\frac{n(1-p)}{n+p}}{k-m}\sum_{\begin{subarray}{c}i_{1},\dots,i_{n-1}\geq
0\\\
i_{1}+2i_{2}+\dots+(n-1)i_{n-1}=m\end{subarray}}c(n,p,i(m))\,\prod\limits_{j=1}^{n-1}{n-1\choose
j}^{i_{j}}$ (49)
$\displaystyle\int\limits_{S^{n-1}_{+}}\left(\sum_{i=1}^{n}\xi_{i}\right)^{m-k+\frac{n(1-p)}{n+p}}\,d\mathcal{H}^{n-1}.$

We look now in more detail at the case $p<-n$. In this case, the exponent
$-\frac{n(n-1)}{n+p}+m$ of $l$ in (49) is strictly positive. We observe that
for $\alpha\in\mathbb{N}\cup\\{0\\}$

$\binom{\frac{n(1-p)}{n+p}}{\alpha}\left\\{\begin{aligned} &=0\quad&\text{ if
}\,\alpha\,&=0\\\ &<0\quad&\text{ if }\,\alpha\,&\text{ is odd }\\\
&>0\quad&\text{ if }\,\alpha\,&\text{ is even}.\end{aligned}\right.$

The analysis of the expression

$F_{m}(p)=\sum\limits_{\begin{subarray}{c}i_{1},\dots,i_{n-1}\geq 0\\\
i_{1}+2i_{2}+\dots+(n-1)i_{n-1}=m\end{subarray}}c(n,p,i(m))\,\prod\limits_{j=1}^{n-1}{n-1\choose
j}^{i_{j}}$

is more involved. We have that

$F_{1}(p)=(n-1)\,\frac{n}{n+p}$

which is increasing and $F_{1}(p)\leq 0$ on $[-\infty,-n)$.

$F_{2}(p)=\frac{(n-1)}{2!}\,\frac{n}{n+p}\left((n-1)\frac{n}{n+p}-1\right),$

which is decreasing and $F_{2}(p)\geq 0$ on $[-\infty,-n)$.

$F_{3}(p)=\frac{(n-1)}{3!}\,\frac{n}{n+p}\left((n-1)^{2}\left(\frac{n}{n+p}\right)^{2}-3(n-1)\frac{n}{n+p}+2\right),$

which is increasing and $F_{3}(p)\leq 0$ on $[-\infty,-n)$.

From the analysis of the signs of $\binom{\frac{n(1-p)}{n+p}}{\alpha}$ and the
$F_{m}$ we conclude from (49) that

$\lim_{l\to\infty}\mathcal{V}_{1}^{p}(K_{l})=-\infty$

and thus $\mathcal{V}_{1}^{p}$ is not lower semi continuous for
$p\in[-\infty,-n)$,

$\lim_{l\to\infty}\mathcal{V}_{2}^{p}(K_{l})=\infty$

and thus $\mathcal{V}_{2}^{p}$ is not upper semi continuous for
$p\in[-\infty,-n)$,

$\lim_{l\to\infty}\mathcal{V}_{3}^{p}(K_{l})=-\infty$

and thus $\mathcal{V}_{3}^{p}$ is not lower semi continuous for
$p\in[-\infty,-n)$. Starting from $m=4$, the behavior of $F_{m}$ is not
monotone anymore on $[-\infty,0)$.

Similarly, for $n<p<n(n-2)$,

$\lim_{l\to\infty}\mathcal{V}_{\frac{n(n-1)}{n+p}}^{p}(K_{l})=F_{\frac{n(n-1)}{n+p}}(p)\int\limits_{S^{n-1}}\left(\sum_{i=1}^{n}\xi_{i}\right)^{\frac{n(1-p)}{n+p}}\,d\mathcal{H}^{n-1}$

is a positive or negative number, which disproves upper or lower semi
continuity, depending on the sign of $F_{\frac{n(n-1)}{n+p}}(p)$.

### Acknowledgement

The second author wants to thank the Hausdorff Research Institute for
Mathematics. Part of the work on the paper was carried out during her stay
there. The authors also wish to thank the Institute for Computational and
Experimental Research in Mathematics (ICERM) for the hospitality. The
manuscript was completed during their participation at the program “Harmonic
Analysis and Convexity”.

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Kateryna Tatarko,  Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1. e-mail<EMAIL_ADDRESS>|  Elisabeth M. Werner,  Department of Mathematics, Case Western Reserve University, Cleveland, Ohio, USA, 44106. e-mail<EMAIL_ADDRESS>
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