# A general functional version of Grünbaum’s inequality

David Alonso-Gutiérrez  Área de análisis matemático, Departamento de
matemáticas, Facultad de Ciencias, Universidad de Zaragoza, C/ Pedro Cerbuna
12, 50009 Zaragoza (Spain), IUMA<EMAIL_ADDRESS>,  Francisco Marín Sola
Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100
Murcia, Spain<EMAIL_ADDRESS><EMAIL_ADDRESS>,  Javier Martín Goñi
Área de análisis matemático, Departamento de matemáticas, Facultad de
Ciencias, Universidad de Zaragoza, C/ Pedro Cerbuna 12, 50009 Zaragoza
(Spain), IUMA and Faculty of Computer Science and Mathematics, University of
Passau, Innstrasse 33, 94032 Passau, Germany<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Jesús Yepes Nicolás

###### Abstract.

A classical inequality by Grünbaum provides a sharp lower bound for the ratio
$\mathrm{vol}(K^{-})/\mathrm{vol}(K)$, where $K^{-}$ denotes the intersection
of a convex body with non-empty interior $K\subset\mathbb{R}^{n}$ with a
halfspace bounded by a hyperplane $H$ passing through the centroid
$\mathrm{g}(K)$ of $K$.

In this paper we extend this result to the case in which the hyperplane $H$
passes by any of the points lying in a whole uniparametric family of
$r$-powered centroids associated to $K$ (depending on a real parameter $r\geq
0$), by proving a more general functional result on concave functions.

The latter result further connects (and allows one to recover) various
inequalities involving the centroid, such as a classical inequality (due to
Minkowski and Radon) that relates the distance of $\mathrm{g}(K)$ to a
supporting hyperplane of $K$, or a result for volume sections of convex bodies
proven independently by Makai Jr. & Martini and Fradelizi.

###### Key words and phrases:

Grünbaum’s inequality, centroid, sections of convex bodies, $p$-concavity

###### 2010 Mathematics Subject Classification:

Primary 26B25, 26D15, 52A38, 52A40; Secondary 52A20

The first author is supported by MICINN project PID2022-137294NB-I00 and DGA
project E48$\\_$23R. The second and the fourth authors are supported by the
grant PID2021-124157NB-I00, funded by MCIN/AEI/10.13039/501100011033/“ERDF A
way of making Europe”, as well as by the grant “Proyecto financiado por la
CARM a través de la convocatoria de Ayudas a proyectos para el desarrollo de
investigación científica y técnica por grupos competitivos, incluida en el
Programa Regional de Fomento de la Investigación Científica y Técnica (Plan de
Actuación 2022) de la Fundación Séneca-Agencia de Ciencia y Tecnología de la
Región de Murcia, REF. 21899/PI/22”. The third author is supported by the
Austrian Science Fund (FWF) Project P32405 Asymptotic Geometric Analysis and
Applications. The fourth author is also supported by the grant
RYC2021-034858-I funded by MCIN/AEI/10.13039/501100011033 and by the “European
Union NextGenerationEU/PRTR”

##  1\. Introduction

Let $K\subset\mathbb{R}^{n}$ be a compact set with positive volume
$\mathrm{vol}(K)$, i.e., with positive $n$-dimensional Lebesgue measure (along
the paper, the $k$-dimensional Lebesgue measure of $M$, provided that $M$ is
measurable, is denoted by $\mathrm{vol}_{k}(M)$ and we will omit the index $k$
when it is equal to the dimension $n$ of the ambient space; furthermore, when
integrating $\mathrm{d}x$ will stand for $\mathrm{d}\mathrm{vol}(x)$). The
centroid of $K$ is the affine-covariant point

$\mathrm{g}(K):=\frac{1}{\mathrm{vol}(K)}\int_{K}x\,\mathrm{d}x.$

According to a classical result by Grünbaum [9], if $K$ is convex (from now on
a compact convex set will be referred to as a _convex body_) with centroid at
the origin, then

(1.1)
$\frac{\mathrm{vol}(K^{-})}{\mathrm{vol}(K)}\geq\left(\frac{n}{n+1}\right)^{n},$

where $K^{-}=K\cap\\{x\in\mathbb{R}^{n}:\left\langle x,u\right\rangle\leq
0\\}$ and $K^{+}=K\cap\\{x\in\mathbb{R}^{n}:\left\langle x,u\right\rangle\geq
0\\}$ represent the parts of $K$ which are split by the (vector) hyperplane
$H=\\{x\in\mathbb{R}^{n}:\left\langle x,u\right\rangle=0\\}$, for any given
$u\in\mathbb{S}^{n-1}$.

There exists a classical inequality similar in spirit to Grünbaum’s result,
attributed to Minkowski for $n=2,3$ and Radon for general $n$, which bounds
the distance from $\mathrm{g}(K)$ to a supporting hyperplane of the convex
body $K$ (see [1, p. 57-58]). This result asserts that when $K$ has centroid
at the origin then $K\subset-nK$, a fact that is equivalent to the following
statement (here $M|E$ denotes the orthogonal projection of the subset
$M\subset\mathbb{R}^{n}$ onto a vector subspace $E$ of $\mathbb{R}^{n}$):

######  Theorem A.

Let $K\subset\mathbb{R}^{n}$ be a convex body with non-empty interior and let
$H$ be a hyperplane. If $K$ has centroid at the origin then

(1.2)
$\frac{\mathrm{vol}_{1}\bigl{(}K^{-}|H^{\perp}\bigr{)}}{\mathrm{vol}_{1}\bigl{(}K|H^{\perp}\bigr{)}}\geq\frac{1}{n+1}.$

Another inequality of this type, but now involving volume sections instead of
projections, is the following inequality (1.3). It was shown (independently)
by [11], and later by [6], who further proved this result when considering
sections by planes of arbitrary dimension.

######  Theorem B ([6, 11]).

Let $K\subset\mathbb{R}^{n}$ be a convex body with non-empty interior, let $H$
be a hyperplane and let $f:[a,b]\longrightarrow\mathbb{R}_{\geq 0}$ be the
function given by $f(t)=\mathrm{vol}_{n-1}\bigl{(}{K\cap(tu+H)}\bigr{)}$. If
$K$ has centroid at the origin then

(1.3) $\frac{f(0)}{\|f\|_{\infty}}\geq\left(\frac{n}{n+1}\right)^{n-1}.$

Grünbaum’s result was extended to the case of sections [7, 15] and projections
[17] of compact convex sets, and generalized to the analytic setting of _log-
concave_ functions [14] (see also [4, Lemma 2.2.6]) and $p$-concave functions
[15], for $p>0$. Other Grünbaum type inequalities involving volumes of
sections of compact convex sets through their centroid, later generalized to
the case of classical and dual _quermassintegrals_ in [16], can be found in
[6, 11].

Exploiting the original proof of Grünbaum in [9], the following extension of
Grünbaum’s inequality (1.1) to the case of compact sets with some concavity
for the function which gives the volumes of cross-sections parallel to a given
hyperplane (having Grünbaum’s inequality as the particular case $p=1/(n-1)$)
was shown in [12].

######  Theorem C ([12]).

Let $K\subset\mathbb{R}^{n}$ be a compact set with non-empty interior and with
centroid at the origin. Let $H$ be a hyperplane such that the function
$f:H^{\bot}\longrightarrow\mathbb{R}_{\geq 0}$ given by
$f(x)=\mathrm{vol}_{n-1}\bigl{(}K\cap(x+H)\bigr{)}$ is $p$-concave, for some
$p\in(0,\infty)$. Then

(1.4)
$\frac{\mathrm{vol}(K^{-})}{\mathrm{vol}(K)}\geq\left(\frac{p+1}{2p+1}\right)^{(p+1)/p}.$

The inequality is sharp.

Following the idea of the proof of the previous result, we will first show
that an analogous statement holds true when one replaces the centroid by the
midpoint in a direction $u\in\mathbb{S}^{n-1}$, namely, the point
$\bigl{[}(a+b)/2\bigr{]}\cdot u$, where $[a,b]$ is the support of the function
$f:\mathbb{R}\longrightarrow\mathbb{R}_{\geq 0}$ given by

$f(t)=\mathrm{vol}_{n-1}\bigl{(}K\cap(tu+H)\bigr{)}$

(see Proposition 2.1).

In view of these results (Theorem C and Proposition 2.1), here we ask about
the possibility of finding other particular points that ensure a large enough
amount of mass in both subsets that are obtained when cutting the given
compact set $K\subset\mathbb{R}^{n}$ by a hyperplane passing through them. To
figure out such a possible family of points we notice that, fixed a unit
direction $u\in\mathbb{S}^{n-1}$, the corresponding components w.r.t. $u$ of
both the centroid and the midpoint have a similar nature. Indeed, the
component of $\mathrm{g}(K)$ w.r.t. $u$ is given by (see (2.2))

$[\mathrm{g}(K)]_{1}=\frac{1}{\mathrm{vol}(K)}\int_{a}^{b}tf(t)\,\mathrm{d}t=\frac{\int_{a}^{b}tf^{1}(t)\,\mathrm{d}t}{\int_{a}^{b}f^{1}(t)\,\mathrm{d}t},$

whereas the corresponding component of the midpoint is

$\frac{a+b}{2}=\frac{\int_{a}^{b}tf^{0}(t)\,\mathrm{d}t}{\int_{a}^{b}f^{0}(t)\,\mathrm{d}t}.$

Thus, with the above-mentioned aim in mind, it seems reasonable to consider
the points $\mathrm{g}_{r}\cdot u$, where

(1.5)
$\mathrm{g}_{r}:=\frac{\int_{a}^{b}tf^{r}(t)\,\mathrm{d}t}{\int_{a}^{b}f^{r}(t)\,\mathrm{d}t}$

for any $r\geq 0$. Here we show that such a uniparametric class of points
allows us to extend Grünbaum’s inequality (or more generally Theorem C) to the
case in which one replaces the classical centroid by any of them.

######  Theorem 1.1.

Let $r\in[0,\infty)$ and let $K\subset\mathbb{R}^{n}$ be a compact set with
non-empty interior having the point $\mathrm{g}_{r}\cdot u$, with respect to
some direction $u\in\mathbb{S}^{n-1}$, at the origin. Let
$H=\\{x\in\mathbb{R}^{n}:\left\langle x,u\right\rangle=0\\}$ be the hyperplane
with normal vector $u$ and assume that the function
$f:H^{\bot}\longrightarrow\mathbb{R}_{\geq 0}$ given by
$f(x)=\mathrm{vol}_{n-1}\bigl{(}K\cap(x+H)\bigr{)}$ is $p$-concave, for some
$p\in(0,\infty)$. If $r\geq 1$ then

$\frac{\mathrm{vol}(K^{-})}{\mathrm{vol}(K)}\geq\left(\frac{p+1}{2p+r}\right)^{(p+1)/p},$

whereas if $0\leq r\leq 1$ then

$\frac{\mathrm{vol}(K^{-})}{\mathrm{vol}(K)}\geq\left(\frac{p+r}{2p+r}\right)^{(p+1)/p}.$

Notice that the cases $r=1$ and $r=0$ correspond to Theorem C and Proposition
2.1, respectively.

Taking into account that, once a unit direction $u\in\mathbb{S}^{n-1}$ is
fixed, the above geometric results are reduced to the study of one variable
functions with certain concavity, here we deal with the corresponding
functional counterpart of these statements (from which the latter result will
be obtained as a consequence of such an equivalent functional one). To this
aim, first we need to define the notion of functional $\alpha$-centroid: given
a non-negative function $h:[a,b]\longrightarrow[0,\infty)$ with positive
integral, for any $\alpha>0$ we will write

(1.6)
$\mathrm{g}_{\alpha}(h):=\frac{\int_{a}^{b}th^{\alpha}(t)\,\mathrm{d}t}{\int_{a}^{b}h^{\alpha}(t)\,\mathrm{d}t}.$

Now, the statement of our main result reads as follows.

######  Theorem 1.2.

Let $h:[a,b]\longrightarrow[0,\infty)$ be a non-negative concave function, and
let $\alpha,\beta>0$. If $\beta\leq\alpha$ then

(1.7)
$\frac{\int_{\mathrm{g}_{\alpha}(h)}^{b}h^{\beta}(t)\,\mathrm{d}t}{\int_{a}^{b}h^{\beta}(t)\,\mathrm{d}t}\geq\left(\frac{\beta+1}{\alpha+2}\right)^{\beta+1},$

whereas if $\alpha\leq\beta$ then

(1.8)
$\frac{\int_{\mathrm{g}_{\alpha}(h)}^{b}h^{\beta}(t)\,\mathrm{d}t}{\int_{a}^{b}h^{\beta}(t)\,\mathrm{d}t}\geq\left(\frac{\alpha+1}{\alpha+2}\right)^{\beta+1}.$

######  Remark 1.1.

We observe that Theorem 1.1 is directly obtained from the previous result by
just taking $h=f^{p}$, $\beta=1/p$ and $\alpha=r\beta$ (where the case $r=0$
is derived when doing $\alpha\to 0^{+}$). Moreover, Theorem 1.2 can be shown
from Theorem 1.1 by just considering the set of revolution $K$ associated to
the radius function $r=(1/\kappa_{n-1})f^{1/(n-1)}$, where $\kappa_{n-1}$ is
the $(n-1)$-dimensional volume of the Euclidean unit ball in
$\mathbb{R}^{n-1}$, $f=h^{\beta}$, $p=1/\beta$ and $r=\alpha/\beta$.
Therefore, in fact, both results (Theorems 1.1 and 1.2) are equivalent.

We would also like to point out that, apart from the already mentioned Theorem
1.1, and thus in particular Theorem C and Grünbaum’s inequality, both Theorems
A and B can be derived as direct applications of Theorem 1.2.

Indeed, on the one hand, applying Theorem 1.2 with $h=f^{1/(n-1)}$, which is
concave because of _Brunn’s concavity principle_ (see e.g. [4, Section 1.2.1]
and also [13, Theorem 12.2.1]), and taking $\alpha=n-1$ and $\beta\to 0^{+}$,
one gets $b/(b-a)\geq 1/(n+1)$, which is exactly (1.2).

On the other hand, applying Theorem 1.2 with $h=f^{1/(n-1)}$ and $\alpha=n-1$,
and then raising both sides of (1.8) to the power $1/\beta$ and taking
$\beta\to\infty$, one has

(1.9)
$\left(\frac{\max_{t\in[0,b]}f(t)}{\|f\|_{\infty}}\right)^{1/(n-1)}\geq\frac{n}{n+1}.$

Since we may assume without loss of generality that
$\|f\|_{\infty}>\max_{t\in[0,b]}f(t)$ (considering otherwise the function
$t\mapsto f(-t)$), we then get $\max_{t\in[0,b]}f(t)=f(0)$ (since $f$ is
$(1/(n-1))$-concave and thus quasi-concave), and therefore (1.9) is nothing
but (1.3).

The paper is organized as follows. In Section 2 we collect some preliminaries
and background, and we show some first results, such as Propositions 2.1 and
2.2. The remaining section of the paper, Section 3, is devoted to the proof of
Theorem 1.2, which is also divided into another two subsections, one for each
of both cases that are distinguished in Theorem 1.2 (namely, $\beta<\alpha$
and $\alpha<\beta$).

##  2\. Preliminaries and first results

We recall that a function
$\varphi:\mathbb{R}^{n}\longrightarrow\mathbb{R}_{\geq 0}$ is $p$-concave, for
$p\in\mathbb{R}\cup\\{\pm\infty\\}$, if

$\varphi\bigl{(}(1-\lambda)x+\lambda
y\bigr{)}\geq\bigl{(}(1-\lambda)\varphi(x)^{p}+\lambda\varphi(y)^{p}\bigr{)}^{1/p}$

for all $x,y\in\mathbb{R}^{n}$ such that $\varphi(x)\varphi(y)>0$ and any
$\lambda\in(0,1)$, where the cases $p=0$, $p=\infty$ and $p=-\infty$ must be
understood as the corresponding expressions that are obtained by continuity,
namely, the geometric mean, the maximum and the minimum (of $\varphi(x)$ and
$\varphi(y)$), respectively. Note that if $p>0$, then $\varphi$ is $p$-concave
if and only if $\varphi^{p}$ is concave on its support
$\\{x\in\mathbb{R}^{n}:\varphi(x)>0\\}$ and thus, in particular, $1$-concave
is just concave (on its support) in the usual sense. A $0$-concave function is
usually called _log-concave_ whereas a $(-\infty)$-concave function is
referred to as _quasi-concave_. Moreover, Jensen’s inequality for means (see
e.g. [10, Section 2.9] and [5, Theorem 1 p. 203]) implies that a $q$-concave
function is also $p$-concave, whenever $q>p$.

For the sake of simplicity, in the following we consider
$H=\\{x\in\mathbb{R}^{n}:\left\langle x,u\right\rangle=0\\}$, for a given
direction $u\in\mathbb{S}^{n-1}$ that we extend to an orthonormal basis
$(u_{1},u_{2},\dots,u_{n})$ of $\mathbb{R}^{n}$, with $u_{1}=u$. Given a set
$M\subset\mathbb{R}^{n}$, and a real number $c\in\mathbb{R}$, we will write
$M^{-}(u,c)=M\cap\\{x\in\mathbb{R}^{n}:\left\langle x,u\right\rangle\leq c\\}$
and $M^{+}(u,c)=M\cap\\{x\in\mathbb{R}^{n}:\left\langle x,u\right\rangle\geq
c\\}$. When dealing with the above-mentioned prescribed hyperplane $H$ we will
just write $H^{-}$ and $H^{+}$ for the corresponding halfspaces determined by
$H$. Moreover, for a compact set with non-empty interior
$K\subset\mathbb{R}^{n}$, we denote by $K(t)=K\cap(tu+H)$ for any
$t\in\mathbb{R}$. We notice that, if $K|H^{\bot}\subset[au,bu]$, Fubini’s
theorem applied to the function $f:\mathbb{R}\longrightarrow\mathbb{R}_{\geq
0}$ given by

$f(t)=\mathrm{vol}_{n-1}\bigl{(}K\cap(tu+H)\bigr{)}$

yields (provided that $a\leq 0$)

(2.1)
$\mathrm{vol}(K)=\int_{a}^{b}f(t)\,\mathrm{d}t\quad\text{and}\quad\mathrm{vol}(K^{-})=\int_{a}^{0}f(t)\,\mathrm{d}t,$

where, as usual, we are identifying the linear subspace spanned by $u$ with
$\mathbb{R}$. Since the set $\\{t\in\mathbb{R}:f(t)>0\\}$ is convex whenever
$f$ is quasi-concave, from now on we will assume, without loss of generality,
that $f(t)>0$ for all $t\in(a,b)$. Furthermore, by Fubini’s theorem, we get

(2.2)
$[\mathrm{g}(K)]_{1}=\frac{1}{\mathrm{vol}(K)}\int_{a}^{b}tf(t)\,\mathrm{d}t$

and thus, in particular, $a<[\mathrm{g}(K)]_{1}<b$ (cf. (2.1)).

It seems natural to wonder about the possibility of getting an analogue of
Theorem C by considering the midpoint in the direction $u$ of $K$ instead its
centroid, that is, the point $\bigl{[}(a+b)/2\bigr{]}\cdot u$. This is the
content of the following result.

######  Proposition 2.1.

Let $K\subset\mathbb{R}^{n}$ be a compact set with non-empty interior and with
midpoint, with respect to some direction $u\in\mathbb{S}^{n-1}$, at the
origin. Let $H=\\{x\in\mathbb{R}^{n}:\left\langle x,u\right\rangle=0\\}$ be
the hyperplane with normal vector $u$ and assume that the function
$f:H^{\bot}\longrightarrow\mathbb{R}_{\geq 0}$ given by
$f(x)=\mathrm{vol}_{n-1}\bigl{(}K\cap(x+H)\bigr{)}$ is $p$-concave, for some
$p\in(0,\infty)$. Then

$\frac{\mathrm{vol}(K^{-})}{\mathrm{vol}(K)}\geq\left(\frac{1}{2}\right)^{(p+1)/p}.$

###### Proof.

In the proof of [12, Theorem 1.1] it is shown that there exists a $p$-affine
function $g_{p}:[-\gamma,\delta]\longrightarrow\mathbb{R}_{\geq 0}$ given by
$g_{p}(t)=c(t+\gamma)^{1/p}$, for some $\gamma,\delta,c>0$, such that
$g_{p}(0)=f(0)$,

$\int_{-\gamma}^{0}g_{p}(t)\,\mathrm{d}t=\int_{a}^{0}f(t)\,\mathrm{d}t\quad\text{and}\quad\int_{0}^{\delta}g_{p}(t)\,\mathrm{d}t=\int_{0}^{b}f(t)\,\mathrm{d}t,$

and further that $-\gamma\leq a<0<\delta\leq b$.

Here, since $K$ has its midpoint (w.r.t. $u$) at the origin, we have that
$a=-b$ and then we get $-\gamma+\delta\leq 0$. Thus

$\frac{\mathrm{vol}(K^{-})}{\mathrm{vol}(K)}=\frac{\int_{-\gamma}^{0}g_{p}(t)\,\mathrm{d}t}{\int_{-\gamma}^{\delta}g_{p}(t)\,\mathrm{d}t}\geq\frac{\int_{-\gamma}^{(-\gamma+\delta)/2}g_{p}(t)\,\mathrm{d}t}{\int_{-\gamma}^{\delta}g_{p}(t)\,\mathrm{d}t}=\left(\frac{1}{2}\right)^{(p+1)/p},$

as desired. ∎

We collect here the following result, originally proved in [2] and [3] (see
also [8] for a detailed presentation), which can be regarded as the functional
counterpart of the _Brunn-Minkowski inequality_.

######  Theorem D (The Borell-Brascamp-Lieb inequality).

Let $\lambda\in(0,1)$. Let $-1/n\leq p\leq\infty$ and let
$f,g,h:\mathbb{R}^{n}\longrightarrow\mathbb{R}_{\geq 0}$ be measurable
functions with positive integrals such that

$h((1-\lambda)x+\lambda y)\geq\bigl{(}(1-\lambda)f(x)^{p}+\lambda
g(y)^{p}\bigr{)}^{1/p}$

for all $x,y\in\mathbb{R}^{n}$ with $f(x)g(y)>0$. Then

(2.3)
$\int_{\mathbb{R}^{n}}h(x)\,\mathrm{d}x\geq\left[(1-\lambda)\left(\int_{\mathbb{R}^{n}}f(x)\,\mathrm{d}x\right)^{q}+\lambda\left(\int_{\mathbb{R}^{n}}g(x)\,\mathrm{d}x\right)^{q}\right]^{1/q},$

where $q=p/(np+1)$.

Now, considering the points $\mathrm{g}_{r}\cdot u$, for any $r>0$, where
$\mathrm{g}_{r}$ is given by (1.5), and following the same idea as in [6,
Theorem 3], we can get a first result concerning this family of points. The
statement reads as follows.

######  Corollary 2.1.

Let $r\in(0,\infty)$ and let $K\subset\mathbb{R}^{n}$ be a compact set with
non-empty interior having the point $\mathrm{g}_{r}\cdot u$, with respect to
some direction $u\in\mathbb{S}^{n-1}$, at the origin. Let
$H=\\{x\in\mathbb{R}^{n}:\left\langle x,u\right\rangle=0\\}$ be the hyperplane
with normal vector $u$ and assume that the function
$f:\mathbb{R}\longrightarrow\mathbb{R}_{\geq 0}$ given by
$f(t)=\mathrm{vol}_{n-1}\bigl{(}K\cap(tu+H)\bigr{)}$ is $p$-concave, for some
$p\in(0,\infty)$. Then

(2.4)
$\frac{\mathrm{vol}(K^{-})}{\mathrm{vol}(K)}\geq\left(\frac{p}{2p+r}\right)^{(p+1)/p}.$

We will derive Corollary 2.1 as a simple application of the following
(slightly more general) functional result.

######  Proposition 2.2.

Let $K\subset\mathbb{R}^{n}$ be a convex body. Let
$g:K\longrightarrow\mathbb{R}_{\geq 0}$ be a concave function and let
$f:K\longrightarrow\mathbb{R}_{\geq 0}$ be a $p$-concave function, with $p>0$.
Then

$g\left(\frac{\int_{K}xf(x)\,\mathrm{d}x}{\int_{K}f(x)\,\mathrm{d}x}\right)\geq\left(\frac{p}{(n+1)p+1}\right)\|g\|_{\infty}.$

###### Proof.

Let $\mu$ be the probability measure whose density function is given by

$\mathrm{d}\mu(x)=\frac{f(x)}{\int_{K}f(x)\,\mathrm{d}x}\mathrm{d}x.$

Since $g$ is concave, using Jensen’s inequality we get that

$\begin{split}g\left(\frac{\int_{K}xf(x)\,\mathrm{d}x}{\int_{K}f(x)\,\mathrm{d}x}\right)=g\left(\int_{K}x\,\mathrm{d}\mu(x)\right)&\geq\int_{K}g(x)\,\mathrm{d}\mu(x)\\\
&=\int_{0}^{\|g\|_{\infty}}\mu\bigl{(}\\{x\in K:g(x)\geq
t\\}\bigr{)}\,\mathrm{d}t,\end{split}$

where in the last identity we have used Fubini’s theorem. Now, since the
density of $\mu$, with respect to the Lebesgue measure, is $p$-concave, from
the Borell-Brascamp-Lieb inequality (2.3) we have that the function
$\varphi(t):=\mu\bigl{(}\\{x\in K:g(x)\geq t\\}\bigr{)}$ is
$\bigl{(}p/(np+1)\bigr{)}$-concave. Indeed, it is enough to apply Theorem D
with the functions

$\frac{f\cdot\chi_{{}_{\\{x\in K\,:\,g(x)\geq
t_{1}\\}}}}{\int_{K}f(x)\,\mathrm{d}x},\quad\frac{f\cdot\chi_{{}_{\\{x\in
K\,:\,g(x)\geq t_{2}\\}}}}{\int_{K}f(x)\,\mathrm{d}x}\quad\text{ and
}\quad\frac{f\cdot\chi_{{}_{\\{x\in K\,:\,g(x)\geq(1-\lambda)t_{1}+\lambda
t_{2}\\}}}}{\int_{K}f(x)\,\mathrm{d}x}$

for any $t_{1},t_{2}\in\mathbb{R}$, where $\chi_{{}_{M}}$ denotes the
characteristic function of the set $M$, one easily obtains the desired
concavity of $\varphi$. Hence, and taking into account that $\varphi(0)=1$ and
$\varphi\bigl{(}\|g\|_{\infty}\bigr{)}\geq 0$, we may assure that
$\varphi(t)^{p/(np+1)}\geq(1-t/\|g\|_{\infty})$ for all
$t\in[0,\|g\|_{\infty}]$. So, by integrating we obtain that

$\begin{split}\int_{0}^{\|g\|_{\infty}}\mu\bigl{(}\\{x\in K:g(x)\geq
t\\}\bigr{)}\,\mathrm{d}t&\geq\int_{0}^{\|g\|_{\infty}}(1-t/\|g\|_{\infty})^{(np+1)/p}\,\mathrm{d}t\\\
&=\left(\frac{p}{(n+1)p+1}\right)\|g\|_{\infty},\end{split}$

from where the result immediately follows. ∎

We conclude this section by showing Corollary 2.1.

###### Proof of Corollary 2.1.

Denoting by $[a,b]$ the support of $f$, let
$\bar{f},g:[a,b]\longrightarrow\mathbb{R}_{\geq 0}$ be the functions given by

$g(t)=\mathrm{vol}\bigl{(}K\cap(tu+H^{-})\bigr{)}^{p/(p+1)}=\left(\int_{a}^{t}f(s)\,\mathrm{d}s\right)^{p/(p+1)}$

and

$\bar{f}(t)=\mathrm{vol}_{n-1}\bigl{(}K\cap(tu+H)\bigr{)}^{r}=f(t)^{r}.$

By hypothesis, it is clear that $\bar{f}$ is $(p/r)$-concave, whereas from the
Borell-Brascamp-Lieb inequality (2.3) we have that $g$ is concave on $[a,b]$.
So, from Proposition 2.2 applied to the functions $\bar{f}$ and $g$, and
taking into account that $\mathrm{g}_{r}=0$, we get that

$\begin{split}\mathrm{vol}(K^{-})=g\left(\frac{\int_{a}^{b}t\bar{f}(t)\,\mathrm{d}t}{\int_{a}^{b}\bar{f}(t)\,\mathrm{d}t}\right)^{(p+1)/p}&\geq\left(\frac{p}{2p+r}\right)^{(p+1)/p}\|g\|_{\infty}^{(p+1)/p}\\\
&=\left(\frac{p}{2p+r}\right)^{(p+1)/p}\mathrm{vol}(K),\end{split}$

as desired. ∎

In the following section, the tighter inequalities collected in Theorem 1.1,
which improve the one that was obtained in Corollary 2.1, will be shown. More
precisely, we will enhance the constant given by (2.4), namely,
$\bigl{(}p/(2p+r)\bigr{)}^{(p+1)/p}$, by other two constants (depending on
whether $r$ is either less than one or greater than one). Furthermore, these
new constants do fit well with (1.4), since, in fact, they all coincide when
$r=1$, that is, when one considers the centroid of the set.

##  3\. Proof of Theorem 1.2

Here we are going to slightly modify the approach followed in [17, Theorem 8]
(which yields our case $\beta=n-1$) to cover all the cases for a general
concave function $h$, which will allow us to show Theorem 1.2. To this aim, we
split the proof into two steps, depending on whether $\beta<\alpha$ or
$\alpha<\beta$. Note also that the case $\alpha=\beta$ is equivalent to the
statement of Theorem C (by just taking $p:=1/\alpha=1/\beta$ and
$f=h^{\alpha}=h^{\beta}$). Before distinguishing whether $\beta<\alpha$ or
$\alpha<\beta$, we make some general considerations.

We may assume, without loss of generality, that $a=0$. Now, let
$L\subset\mathbb{R}^{2}$ be the convex body

$L:=\bigl{\\{}(x,y)\in\mathbb{R}^{2}\,:\,0\leq x\leq b,\,0\leq y\leq
h(x)\bigr{\\}}$

and notice that, from Fubini’s theorem, we have

(3.1)
$\mathrm{g}_{\alpha}(h)=\frac{\int_{0}^{b}th(t)^{\alpha}\,\mathrm{d}t}{\int_{0}^{b}h(t)^{\alpha}\,\mathrm{d}t}=\frac{\int_{L}\langle
x,\mathrm{e}_{1}\rangle\langle
x,\mathrm{e}_{2}\rangle^{\alpha-1}\,\mathrm{d}x}{\int_{L}\langle
x,\mathrm{e}_{2}\rangle^{\alpha-1}\,\mathrm{d}x}.$

Let $\mu_{\beta}$ be the measure on $\mathbb{R}^{2}$ given by
$\,\mathrm{d}\mu_{\beta}(x)=\langle
x,\mathrm{e}_{2}\rangle^{\beta-1}\,\mathrm{d}x$. Then

$\mathrm{g}_{\alpha}(h)=\frac{\int_{L}\langle x,\mathrm{e}_{1}\rangle\langle
x,\mathrm{e}_{2}\rangle^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)}{\int_{L}\langle
x,\mathrm{e}_{2}\rangle^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)}$

and

$\begin{split}\frac{\int_{\mathrm{g}_{\alpha}(h)}^{b}h(t)^{\beta}\,\mathrm{d}t}{\int_{0}^{b}h(t)^{\beta}\,\mathrm{d}t}&=\frac{\int_{\\{x\in
L\,:\,\langle x,\mathrm{e}_{1}\rangle\geq\mathrm{g}_{\alpha}(h)\\}}\langle
x,\mathrm{e}_{2}\rangle^{\beta-1}\,\mathrm{d}x}{\int_{L}\langle
x,\mathrm{e}_{2}\rangle^{\beta-1}\,\mathrm{d}x}\\\ &=\frac{\mu_{\beta}\\{x\in
L\,:\,\langle
x,\mathrm{e}_{1}\rangle\geq\mathrm{g}_{\alpha}(h)\\}}{\mu_{\beta}(L)}.\end{split}$

We may assure that there exist $\gamma<\delta$ and $c>0$ in such a way that
the affine decreasing function $g:[\gamma,\delta]\longrightarrow[0,\infty)$
given by

$g(t)=c(\delta-t)$

satisfies

(3.2)
$\begin{array}[]{l}\text{(i)}\;g\bigl{(}\mathrm{g}_{\alpha}(h)\bigr{)}=h\bigl{(}\mathrm{g}_{\alpha}(h)\bigr{)},\\\\[11.38109pt]
\text{(ii)}\;\int_{\gamma}^{\delta}g(t)^{\beta}\,\mathrm{d}t=\int_{0}^{b}h(t)^{\beta}\,\mathrm{d}t,\quad\text{
and }\\\\[11.38109pt]
\text{(iii)}\;\int_{\mathrm{g}_{\alpha}(h)}^{\delta}g(t)^{\beta}\,\mathrm{d}t=\int_{\mathrm{g}_{\alpha}(h)}^{b}h(t)^{\beta}\,\mathrm{d}t.\end{array}$

Indeed, taking

$\delta=\frac{\beta+1}{h\bigl{(}\mathrm{g}_{\alpha}(h)\bigr{)}^{\beta}}\int_{\mathrm{g}_{\alpha}(h)}^{b}h(t)^{\beta}\,\mathrm{d}t+\,\mathrm{g}_{\alpha}(h),\quad
c=\frac{h\bigl{(}\mathrm{g}_{\alpha}(h)\bigr{)}}{\delta-\mathrm{g}_{\alpha}(h)}$

and

$\gamma=\delta-\left(\frac{\beta+1}{c^{\beta}}\int_{0}^{b}h(t)^{\beta}\,\mathrm{d}t\right)^{1/(\beta+1)},$

elementary computations show (3.2).

Figure 1. Sets $L$ and $L_{g}$.

Then, denoting by $L_{g}\subset\mathbb{R}^{2}$ the triangle (see Figure 1)
given by

$L_{g}:=\bigl{\\{}(x,y)\in\mathbb{R}^{2}\,:\,\gamma\leq x\leq\delta,\,0\leq
y\leq g(x)\bigr{\\}},$

from (ii) and (iii) in (3.2) (by using Fubini’s theorem), and the relative
position of $h$ and $g$, given by the concavity of $h$ and the relation
$g\bigl{(}\mathrm{g}_{\alpha}(h)\bigr{)}=h\bigl{(}\mathrm{g}_{\alpha}(h)\bigr{)}$
(see Figure 2), we have that

(3.3)
$\begin{array}[]{l}\text{(i)}\;\mu_{\beta}(L)=\mu_{\beta}(L_{g}),\\\\[8.53581pt]
\text{(ii)}\;\mu_{\beta}\bigl{(}\\{x\in L:\langle
x,\mathrm{e}_{1}\rangle\geq\mathrm{g}_{\alpha}(h)\\}\bigr{)}\\!=\\!\mu_{\beta}\bigl{(}\\{x\in
L_{g}:\langle
x,\mathrm{e}_{1}\rangle\geq\mathrm{g}_{\alpha}(h)\\}\bigr{)},\\\\[8.53581pt]
\text{(iii)}\;0\leq\gamma\leq\mathrm{g}_{\alpha}(h)\leq
b\leq\delta.\end{array}$

$g$$h$$\gamma$$\delta$$0$$b$$\mathrm{g}_{\alpha}$ Figure 2. Relative position
of the functions $h$ and $g$.

Moreover, defining $g(t):=0$ for all $t\in[0,\gamma]$ and $h(t):=0$ for any
$t\in[b,\delta]$, there exists $x_{0}\in(\mathrm{g}_{\alpha}(h),b]$ such that
$h(t)\geq g(t)$ for all $t\in[0,\gamma]\cup[\mathrm{g}_{\alpha}(h),x_{0}]$ and
$h(t)\leq g(t)$ otherwise (see Figure 2 -there, observe that $x_{0}$ coincides
with $b$). Hence, on the one hand, for every
$s\in[\mathrm{g}_{\alpha}(h),x_{0}]$ (the case of $s\geq x_{0}$ immediately
follows) we get that

$\begin{split}\int_{s}^{b}h(t)^{\beta}\,\mathrm{d}t&=\int_{\mathrm{g}_{\alpha}(h)}^{b}h(t)^{\beta}\,\mathrm{d}t-\int_{\mathrm{g}_{\alpha}(h)}^{s}h(t)^{\beta}\,\mathrm{d}t\\\
&\leq\int_{\mathrm{g}_{\alpha}(h)}^{\delta}g(t)^{\beta}\,\mathrm{d}t-\int_{\mathrm{g}_{\alpha}(h)}^{s}g(t)^{\beta}\,\mathrm{d}t=\int_{s}^{\delta}g(t)^{\beta}\,\mathrm{d}t.\end{split}$

On the other hand, for every $s\in[\gamma,\mathrm{g}_{\alpha}(h)]$ (again, the
case of $s\leq\gamma$ immediately follows) we have that

$\begin{split}\int_{s}^{b}h(t)^{\beta}\,\mathrm{d}t&=\int_{\mathrm{g}_{\alpha}(h)}^{b}h(t)^{\beta}\,\mathrm{d}t+\int_{s}^{\mathrm{g}_{\alpha}(h)}h(t)^{\beta}\,\mathrm{d}t\\\
&\leq\int_{\mathrm{g}_{\alpha}(h)}^{\delta}g(t)^{\beta}\,\mathrm{d}t+\int_{s}^{\mathrm{g}_{\alpha}(h)}g(t)^{\beta}\,\mathrm{d}t=\int_{s}^{\delta}g(t)^{\beta}\,\mathrm{d}t.\end{split}$

Therefore,

(3.4)
$\begin{split}\mu_{\beta}\bigl{(}L^{+}(\mathrm{e}_{1},s)\bigr{)}&=\mu_{\beta}\bigl{(}\\{x\in
L\,:\,\langle x,\mathrm{e}_{1}\rangle\geq
s\\}\bigr{)}=\int_{s}^{b}h(t)^{\beta}\,\mathrm{d}t\\\
&\leq\int_{s}^{\delta}g(t)^{\beta}\,\mathrm{d}t=\mu_{\beta}\bigl{(}\\{x\in
L_{g}\,:\,\langle x,\mathrm{e}_{1}\rangle\geq
s\\}\bigr{)}=\mu_{\beta}\bigl{(}L_{g}^{+}(\mathrm{e}_{1},s)\bigr{)}\end{split}$

for every $s\in[0,\delta]$, where we are using the notation $L_{g}^{+}(u,c)$
to represent the set $\bigl{(}L_{g}\bigr{)}^{+}(u,c)$, for a given direction
$u\in\mathbb{S}^{n-1}$ and a real number $c\in\mathbb{R}$.

###  3.1. The case of $\beta<\alpha$

We devote this section to proving the first part of Theorem 1.2, namely, we
show (1.7) provided that $\beta<\alpha$.

We will first prove that there exists a non-negative and concave function
$\varphi:[\gamma,\delta]\longrightarrow[0,\|h\|_{\infty}]$ such that

(3.5) $\mathrm{g}_{\alpha}(h)\leq\frac{\int_{L_{g}}\langle
x,\mathrm{e}_{1}\rangle\varphi(\langle
x,\mathrm{e}_{1}\rangle)^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)}{\int_{L_{g}}\varphi(\langle
x,\mathrm{e}_{1}\rangle)^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)}.$

To this aim, we consider the function
$W:[0,\|h\|_{\infty}]\longrightarrow[0,\mu_{\beta}(L)]$ given by

$W(s)=\mu_{\beta}\bigl{(}\\{x\in L\,:\,\langle x,\mathrm{e}_{2}\rangle\geq
s\\}\bigr{)},$

which is clearly both strictly decreasing and surjective. We may then define
the non-negative function $w:[0,\|h\|_{\infty}]\longrightarrow[\gamma,\delta]$
that satisfies

$W(s)=\mu_{\beta}\bigl{(}\\{x\in L_{g}\,:\,\langle x,\mathrm{e}_{1}\rangle\geq
w(s)\\}\bigr{)}$

for any $s\in[0,\|h\|_{\infty}]$. Indeed, since

$W(s)=\frac{c^{\beta}(\delta-w(s))^{\beta+1}}{\beta(\beta+1)},$

we get that

$w(s)=\delta-\left(\frac{\beta(\beta+1)}{c^{\beta}}W(s)\right)^{1/(\beta+1)}.$

Notice that, from the Borell-Brascamp-Lieb inequality (2.3), we have that the
function $W^{1/(\beta+1)}$ is concave (since the density of $\mu_{\beta}$,
with respect to the Lebesgue measure, is
$\bigl{(}1/(\beta-1)\bigr{)}$-concave). Therefore, $w$ is strictly increasing,
surjective and convex, and then there exists the function
$\varphi=w^{-1}:[\gamma,\delta]\longrightarrow[0,\|h\|_{\infty}]$, which is
further (strictly increasing and) concave.

Now, we will start by bounding from above the right-hand side of (3.1). By
using Fubini’s theorem, (iii) in (3.3) and (3.4), we have

(3.6) $\begin{split}\frac{1}{\alpha-\beta}&\int_{L}\langle
x,\mathrm{e}_{1}\rangle\langle
x,\mathrm{e}_{2}\rangle^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)\\\
=&\int_{L}\int_{0}^{\langle
x,\mathrm{e}_{1}\rangle}\,\mathrm{d}s_{1}\int_{0}^{\langle
x,\mathrm{e}_{2}\rangle}s_{2}^{\alpha-\beta-1}\,\mathrm{d}s_{2}\,\mathrm{d}\mu_{\beta}(x)\\\
=&\int_{0}^{b}\int_{0}^{\|h\|_{\infty}}s_{2}^{\alpha-\beta-1}\mu_{\beta}\bigl{(}L^{+}(\mathrm{e}_{1},s_{1})\cap
L^{+}(\mathrm{e}_{2},s_{2})\bigr{)}\,\mathrm{d}s_{2}\,\mathrm{d}s_{1}\\\
\leq&\int_{0}^{b}\int_{0}^{\|h\|_{\infty}}s_{2}^{\alpha-\beta-1}\min\left\\{\mu_{\beta}\bigl{(}L^{+}(\mathrm{e}_{1},s_{1})\bigr{)},\mu_{\beta}\bigl{(}L^{+}(\mathrm{e}_{2},s_{2})\bigr{)}\right\\}\,\mathrm{d}s_{2}\,\mathrm{d}s_{1}\\\
\leq&\int_{0}^{b}\int_{0}^{\|h\|_{\infty}}s_{2}^{\alpha-\beta-1}\min\left\\{\mu_{\beta}\bigl{(}L_{g}^{+}(\mathrm{e}_{1},s_{1})\bigr{)},\mu_{\beta}\bigl{(}L^{+}(\mathrm{e}_{2},s_{2})\bigr{)}\right\\}\,\mathrm{d}s_{2}\,\mathrm{d}s_{1}\\\
\leq&\int_{0}^{\delta}\int_{0}^{\|h\|_{\infty}}s_{2}^{\alpha-\beta-1}\min\left\\{\mu_{\beta}\bigl{(}L_{g}^{+}(\mathrm{e}_{1},s_{1})\bigr{)},\mu_{\beta}\bigl{(}L^{+}(\mathrm{e}_{2},s_{2})\bigr{)}\right\\}\,\mathrm{d}s_{2}\,\mathrm{d}s_{1}.\end{split}$

So, on the one hand, from (3.6) we get

(3.7) $\begin{split}\frac{1}{\alpha-\beta}&\int_{L}\langle
x,\mathrm{e}_{1}\rangle\langle
x,\mathrm{e}_{2}\rangle^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)\\\
\leq&\int_{0}^{\delta}\int_{0}^{\|h\|_{\infty}}s_{2}^{\alpha-\beta-1}\min\left\\{\mu_{\beta}\bigl{(}L_{g}^{+}(\mathrm{e}_{1},s_{1})\bigr{)},\mu_{\beta}\bigl{(}L^{+}(\mathrm{e}_{2},s_{2})\bigr{)}\right\\}\,\mathrm{d}s_{2}\,\mathrm{d}s_{1}\\\
=&\int_{0}^{\delta}\int_{0}^{\|h\|_{\infty}}s_{2}^{\alpha-\beta-1}\min\left\\{\mu_{\beta}\bigl{(}L_{g}^{+}(\mathrm{e}_{1},s_{1})\bigr{)},\mu_{\beta}\Bigl{(}L_{g}^{+}\bigl{(}\mathrm{e}_{1},w(s_{2})\bigr{)}\Bigr{)}\right\\}\,\mathrm{d}s_{2}\,\mathrm{d}s_{1}\\\
=&\int_{0}^{\delta}\int_{0}^{\|h\|_{\infty}}s_{2}^{\alpha-\beta-1}\mu_{\beta}\Bigl{(}L_{g}^{+}(\mathrm{e}_{1},s_{1})\cap
L_{g}^{+}\bigl{(}\mathrm{e}_{1},w(s_{2})\bigr{)}\Bigr{)}\,\mathrm{d}s_{2}\,\mathrm{d}s_{1}\\\
=&\,\frac{1}{\alpha-\beta}\int_{L_{g}}\langle
x,\mathrm{e}_{1}\rangle\varphi(\langle
x,\mathrm{e}_{1}\rangle)^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x),\end{split}$

where in the last equality above we have used that $\langle
x,\mathrm{e}_{1}\rangle\geq w(s_{2})$ if and only if $\varphi(\langle
x,\mathrm{e}_{1}\rangle)\geq s_{2}$.

On the other hand, since $\alpha-\beta>0$, from Fubini’s theorem we have that

(3.8) $\begin{split}\frac{1}{\alpha-\beta}\int_{L}\langle
x,\mathrm{e}_{2}\rangle^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)=&\int_{0}^{\|h\|_{\infty}}s^{\alpha-\beta-1}\mu_{\beta}\bigl{(}\\{x\in
L\,:\,\langle x,\mathrm{e}_{2}\rangle\geq
s\\}\bigr{)}\,\mathrm{d}s\\\\[2.84526pt]
=&\int_{0}^{\|h\|_{\infty}}s^{\alpha-\beta-1}\mu_{\beta}\bigl{(}\\{x\in
L_{g}\,:\,\langle x,\mathrm{e}_{1}\rangle\geq
w(s)\\}\bigr{)}\,\mathrm{d}s\\\\[2.84526pt]
=&\int_{0}^{\|h\|_{\infty}}s^{\alpha-\beta-1}\mu_{\beta}\bigl{(}\\{x\in
L_{g}\,:\,\varphi(\langle x,\mathrm{e}_{1}\rangle)\geq
s\\}\bigr{)}\,\mathrm{d}s\\\\[2.84526pt] =&\int_{L_{g}}\varphi(\langle
x,\mathrm{e}_{1}\rangle)^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x).\end{split}$

Hence, from (3.7) and (3.8) (and using (3.1)) we obtain (3.5), as desired.

Now we will prove that for any concave function
$\varphi:[\gamma,\delta]\longrightarrow[0,\infty)$ we have that

(3.9) $\frac{\int_{L_{g}}\langle x,\mathrm{e}_{1}\rangle\varphi(\langle
x,\mathrm{e}_{1}\rangle)^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)}{\int_{L_{g}}\varphi(\langle
x,\mathrm{e}_{1}\rangle)^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)}\leq\frac{\int_{L_{g}}\langle
x,\mathrm{e}_{1}\rangle\left(\langle
x,\mathrm{e}_{1}\rangle-\gamma\right)^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)}{\int_{L_{g}}\left(\langle
x,\mathrm{e}_{1}\rangle-\gamma\right)^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)}.$

To this aim, let $C_{1}>0$ be such that

(3.10) $\int_{L_{g}}\bigl{(}C_{1}(\langle
x,\mathrm{e}_{1}\rangle-\gamma)\bigr{)}^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)=\int_{L_{g}}\varphi(\langle
x,\mathrm{e}_{1}\rangle)^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x).$

Since the latter identity is equivalent (by Fubini’s theorem) to

$\int_{\gamma}^{\delta}\left(\bigl{(}C_{1}(t-\gamma)\bigr{)}^{\alpha-\beta}-\varphi(t)^{\alpha-\beta}\right)g(t)^{\beta}\,\mathrm{d}t=0,$

we may assert, taking into account that $\varphi$ is concave, that there
exists $t_{0}\in(\gamma,\delta)$ such that

(3.11) $\begin{array}[]{l}\text{(i)}\;C_{1}(t-\gamma)\leq\varphi(t)\text{ for
every }\gamma\leq t\leq t_{0},\quad\text{and}\\\\[17.07164pt]
\text{(ii)}\;C_{1}(t-\gamma)\geq\varphi(t)\text{ for every }t_{0}\leq
t\leq\delta.\end{array}$

Then, from (3.10) and (3.11) (and using Fubini’s theorem), we get

$\begin{split}\beta\biggl{(}\int_{L_{g}}\langle
x,\mathrm{e}_{1}\rangle&\varphi(\langle
x,\mathrm{e}_{1}\rangle)^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)-\int_{L_{g}}\langle
x,\mathrm{e}_{1}\rangle\bigl{(}C_{1}(\langle
x,\mathrm{e}_{1}\rangle-\gamma)\bigr{)}^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)\biggr{)}\\\
=&\int_{\gamma}^{\delta}t\Bigl{(}\varphi(t)^{\alpha-\beta}-\bigl{(}C_{1}(t-\gamma)\bigr{)}^{\alpha-\beta}\Bigr{)}g(t)^{\beta}\,\mathrm{d}t\\\
=&\int_{\gamma}^{t_{0}}t\Bigl{(}\varphi(t)^{\alpha-\beta}-\bigl{(}C_{1}(t-\gamma)\bigr{)}^{\alpha-\beta}\Bigr{)}g(t)^{\beta}\,\mathrm{d}t\\\
+&\int_{t_{0}}^{\delta}t\Bigl{(}\varphi(t)^{\alpha-\beta}-\bigl{(}C_{1}(t-\gamma)\bigr{)}^{\alpha-\beta}\Bigr{)}g(t)^{\beta}\,\mathrm{d}t\\\
\leq&\,t_{0}\int_{\gamma}^{t_{0}}\left(\varphi(t)^{\alpha-\beta}-\bigl{(}C_{1}(t-\gamma)\bigr{)}^{\alpha-\beta}\right)g(t)^{\beta}\,\mathrm{d}t\\\
+&\,t_{0}\int_{t_{0}}^{\delta}\left(\varphi(t)^{\alpha-\beta}-\bigl{(}C_{1}(t-\gamma)\bigr{)}^{\alpha-\beta}\right)g(t)^{\beta}\,\mathrm{d}t\\\
=&\,t_{0}\int_{\gamma}^{\delta}\left(\varphi(t)^{\alpha-\beta}-\bigl{(}C_{1}(t-\gamma)\bigr{)}^{\alpha-\beta}\right)g(t)^{\beta}\,\mathrm{d}t\\\
=&\,\beta t_{0}\int_{L_{g}}\left(\varphi(\langle
x,\mathrm{e}_{1}\rangle)^{\alpha-\beta}-\bigl{(}C_{1}(\langle
x,\mathrm{e}_{1}\rangle-\gamma)\bigr{)}^{\alpha-\beta}\right)\,\mathrm{d}\mu_{\beta}(x)=0.\end{split}$

Thus, we have

$\int_{L_{g}}\langle x,\mathrm{e}_{1}\rangle\varphi(\langle
x,\mathrm{e}_{1}\rangle)^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)\leq\int_{L_{g}}\langle
x,\mathrm{e}_{1}\rangle\bigl{(}C_{1}(\langle
x,\mathrm{e}_{1}\rangle-\gamma)\bigr{)}^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x),$

which, together with (3.10), yields (3.9).

Now, we will compute the right-hand side of (3.9). On the one hand,

$\begin{split}\int_{L_{g}}\left(\langle
x,\mathrm{e}_{1}\rangle-\gamma\right)^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)=&\frac{c^{\beta}}{\beta}\int_{\gamma}^{\delta}(t-\gamma)^{\alpha-\beta}(\delta-t)^{\beta}\,\mathrm{d}t\\\
=&\frac{c^{\beta}}{\beta}(\delta-\gamma)^{\alpha+1}\int_{0}^{1}s^{\alpha-\beta}(1-s)^{\beta}\,\mathrm{d}s\\\
=&\frac{c^{\beta}}{\beta}(\delta-\gamma)^{\alpha+1}\frac{\Gamma\left(\alpha-\beta+1\right)\Gamma\left(\beta+1\right)}{\Gamma\left(\alpha+2\right)},\end{split}$

whereas, on the other hand, we obtain

$\begin{split}&\quad\quad\int_{L_{g}}\langle
x,\mathrm{e}_{1}\rangle\left(\langle
x,\mathrm{e}_{1}\rangle-\gamma\right)^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)=\frac{c^{\beta}}{\beta}\int_{\gamma}^{\delta}t(t-\gamma)^{\alpha-\beta}(\delta-t)^{\beta}\,\mathrm{d}t\\\
&=\gamma\frac{c^{\beta}}{\beta}(\delta-\gamma)^{\alpha+1}\int_{0}^{1}s^{\alpha-\beta}(1-s)^{\beta}\,\mathrm{d}s+\frac{c^{\beta}}{\beta}(\delta-\gamma)^{\alpha+2}\int_{0}^{1}s^{\alpha-\beta+1}(1-s)^{\beta}\,\mathrm{d}s\\\
&=\frac{c^{\beta}}{\beta}(\delta-\gamma)^{\alpha+1}\frac{\Gamma\left(\alpha-\beta+1\right)\Gamma\left(\beta+1\right)}{\Gamma\left(\alpha+2\right)}\left(\gamma+(\delta-\gamma)\frac{\alpha-\beta+1}{\alpha+2}\right).\end{split}$

Hence, we have

$\frac{\int_{L_{g}}\langle x,\mathrm{e}_{1}\rangle\left(\langle
x,\mathrm{e}_{1}\rangle-\gamma\right)^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)}{\int_{L_{g}}\left(\langle
x,\mathrm{e}_{1}\rangle-\gamma\right)^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)}=\gamma+(\delta-\gamma)\frac{\alpha-\beta+1}{\alpha+2},$

and therefore, this together with (3.5) and (3.9) yields

$\mathrm{g}_{\alpha}(h)\leq\gamma+(\delta-\gamma)\frac{\alpha-\beta+1}{\alpha+2}=:g_{0}.$

Finally, the latter relation jointly with (ii) and (iii) in (3.2) gives us

$\begin{split}\frac{\int_{\mathrm{g}_{\alpha}(h)}^{b}h(t)^{\beta}\,\mathrm{d}t}{\int_{a}^{b}h(t)^{\beta}\,\mathrm{d}t}&=\frac{\int_{\mathrm{g}_{\alpha}(h)}^{\delta}g(t)^{\beta}\,\mathrm{d}t}{\int_{\gamma}^{\delta}g(t)^{\beta}\,\mathrm{d}t}\geq\frac{\int_{g_{0}}^{\delta}g(t)^{\beta}\,\mathrm{d}t}{\int_{\gamma}^{\delta}g(t)^{\beta}\,\mathrm{d}t}=\left(\frac{\delta-
g_{0}}{\delta-\gamma}\right)^{\beta+1}\\\
&=\left(1-\frac{\alpha-\beta+1}{\alpha+2}\right)^{\beta+1}=\left(\frac{\beta+1}{\alpha+2}\right)^{\beta+1},\end{split}$

as desired. This finishes the proof of (1.7).

###  3.2. The case of $\alpha<\beta$

Now we show the second part of Theorem 1.2, namely, (1.8) provided that
$\alpha<\beta$. We point out that here we use an approach similar to the one
followed in Subsection 3.1, but with the main difference that we need to
truncate the sets $L$ and $L_{g}$ due to certain integrability issues, since
now the exponent of some functions under the integral sign (vanishing at some
points of the domains of integration) is $\alpha-\beta<0$.

We start by considering the function
$W_{1}:[0,\|h\|_{\infty}]\longrightarrow[0,\mu_{\beta}(L)]$ given by

$W_{1}(s)=\mu_{\beta}\bigl{(}\\{x\in L\,:\,\langle x,\mathrm{e}_{2}\rangle\leq
s\\}\bigr{)},$

which is clearly both strictly increasing and surjective. We may then define
the function $w_{1}:[0,\|h\|_{\infty}]\longrightarrow[\gamma,\delta]$ that
satisfies

$W_{1}(s)=\mu_{\beta}\bigl{(}\\{x\in L_{g}\,:\,\langle
x,\mathrm{e}_{1}\rangle\geq w_{1}(s)\\}\bigr{)}$

for any $s\in[0,\|h\|_{\infty}]$. Indeed, since

$W_{1}(s)=\frac{c^{\beta}\bigl{(}\delta-
w_{1}(s)\bigr{)}^{\beta+1}}{\beta(\beta+1)},$

we get that

$w_{1}(s)=\delta-\left(\frac{\beta(\beta+1)}{c^{\beta}}W_{1}(s)\right)^{1/(\beta+1)}.$

Note that, from the Borell-Brascamp-Lieb inequality (2.3), we have that the
function $W_{1}^{1/(\beta+1)}$ is concave (since the density of $\mu_{\beta}$,
with respect to the Lebesgue measure, is
$\bigl{(}1/(\beta-1)\bigr{)}$-concave). Therefore, $w_{1}$ is strictly
decreasing, surjective and convex, and then there exists the function
$\varphi_{1}=w_{1}^{-1}:[\gamma,\delta]\longrightarrow[0,\|h\|_{\infty}]$,
which is further (strictly decreasing and) concave.

Now, for any $0<\varepsilon\leq\|h\|_{\infty}$ we define the sets

$L_{\varepsilon}:=\\{x\in L\,:\langle
x,\mathrm{e}_{2}\rangle\geq\varepsilon\\}$

and

$L_{g,\varepsilon}:=\\{x\in L_{g}\,:\langle x,\mathrm{e}_{1}\rangle\leq
w_{1}(\varepsilon)\\}.$

Figure 3. Sets $L_{\varepsilon}$ and $L_{g,\varepsilon}$.

Notice that, from the definition of $W_{1}$ and $w_{1}$ (jointly with (i) in
(3.3)) we have that

(3.12)
$\begin{array}[]{l}\text{(i)}\;\mu_{\beta}(L_{\varepsilon})=\mu_{\beta}(L_{g,\varepsilon}),\quad\text{and}\\\\[11.38109pt]
\text{(ii)}\;\mu_{\beta}\bigl{(}\\{x\in L_{\varepsilon}\,:\langle
x,\mathrm{e}_{2}\rangle\leq s\\}\bigr{)}\\!=\\!\mu_{\beta}\bigl{(}\\{x\in
L_{g,\varepsilon}\,:\langle x,\mathrm{e}_{1}\rangle\geq
w_{1}(s)\\}\bigr{)},\\\\[11.38109pt] \text{for all }\varepsilon\leq
s\leq\|h\|_{\infty}.\end{array}$

We will first prove that, for any $0<\varepsilon\leq\|h\|_{\infty}$, we have

(3.13) $\begin{split}\frac{\int_{L_{\varepsilon}}\langle
x,\mathrm{e}_{1}\rangle\langle
x,\mathrm{e}_{2}\rangle^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)}{\int_{L_{\varepsilon}}\langle
x,\mathrm{e}_{2}\rangle^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)}&<\frac{\int_{L_{g,\varepsilon}}\langle
x,\mathrm{e}_{1}\rangle\varphi_{1}(\langle
x,\mathrm{e}_{1}\rangle)^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)}{\int_{L_{g,\varepsilon}}\varphi_{1}(\langle
x,\mathrm{e}_{1}\rangle)^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)}\\\\[5.69054pt]
&+\frac{b\delta\varepsilon^{\alpha}}{\int_{L_{\varepsilon}}\langle
x,\mathrm{e}_{2}\rangle^{\alpha}\,\mathrm{d}x}.\end{split}$

To this aim, we will observe that, denoting by
$b_{\varepsilon}=\max\bigl{\\{}x\in\mathbb{R}:(x,y)\in
L_{\varepsilon}\bigr{\\}}$, for any $0\leq s\leq b_{\varepsilon}$ we have that

(3.14) $\mu_{\beta}\bigl{(}\\{x\in L_{\varepsilon}\,:\,\langle
x,\mathrm{e}_{1}\rangle\geq s\\}\bigr{)}\leq\mu_{\beta}\bigl{(}\\{x\in
L_{g,\varepsilon}\,:\,\langle x,\mathrm{e}_{1}\rangle\geq
s\\}\bigr{)}+W_{1}(\varepsilon).$

Indeed, taking into account that if $s\leq w_{1}(\varepsilon)$ then

$\mu_{\beta}\bigl{(}\\{x\in L_{g}\,:\,\langle x,\mathrm{e}_{1}\rangle\geq
s\\}\bigr{)}=\mu_{\beta}\bigl{(}\\{x\in L_{g,\varepsilon}\,:\,\langle
x,\mathrm{e}_{1}\rangle\geq s\\}\bigr{)}+W_{1}(\varepsilon),$

and if $s>w_{1}(\varepsilon)$ then

$\mu_{\beta}\bigl{(}\\{x\in L_{g}\,:\,\langle x,\mathrm{e}_{1}\rangle\geq
s\\}\bigr{)}\leq W_{1}(\varepsilon),$

we get, from (3.4), that

$\begin{split}\mu_{\beta}\bigl{(}\\{x\in L_{\varepsilon}\,:\,\langle
x,\mathrm{e}_{1}\rangle\geq s\\}\bigr{)}&\leq\mu_{\beta}\bigl{(}\\{x\in
L\,:\,\langle x,\mathrm{e}_{1}\rangle\geq s\\}\bigr{)}\\\
&\leq\mu_{\beta}\bigl{(}\\{x\in L_{g}\,:\,\langle x,\mathrm{e}_{1}\rangle\geq
s\\}\bigr{)}\\\ &\leq\mu_{\beta}\bigl{(}\\{x\in L_{g,\varepsilon}\,:\,\langle
x,\mathrm{e}_{1}\rangle\geq s\\}\bigr{)}+W_{1}(\varepsilon)\end{split}$

for all $0\leq s\leq b_{\varepsilon}$, which shows (3.14). This, together with

$W_{1}(\varepsilon)\leq\int_{0}^{b}\int_{0}^{\varepsilon}y^{\beta-1}\,\mathrm{d}y\,\mathrm{d}x=\frac{b}{\beta}\varepsilon^{\beta}$

and (ii) in (3.12), implies that

$\begin{split}\min&\left\\{\mu_{\beta}\bigl{(}L_{\varepsilon}^{+}(\mathrm{e}_{1},s_{1})\bigr{)},\mu_{\beta}\bigl{(}L_{\varepsilon}^{-}(\mathrm{e}_{2},1/s_{2})\bigr{)}\right\\}\\\
&\leq\min\left\\{\mu_{\beta}\bigl{(}L_{g,\varepsilon}^{+}(\mathrm{e}_{1},s_{1})\bigr{)}+W_{1}(\varepsilon),\mu_{\beta}\Bigl{(}L_{g,\varepsilon}^{+}\bigl{(}\mathrm{e}_{1},w_{1}(1/s_{2})\bigr{)}\Bigr{)}\right\\}\\\
&\leq
W_{1}(\varepsilon)+\min\left\\{\mu_{\beta}\bigl{(}L_{g,\varepsilon}^{+}(\mathrm{e}_{1},s_{1})\bigr{)},\mu_{\beta}\Bigl{(}L_{g,\varepsilon}^{+}\bigl{(}\mathrm{e}_{1},w_{1}(1/s_{2})\bigr{)}\Bigr{)}\right\\}\\\
&\leq\frac{b}{\beta}\varepsilon^{\beta}+\min\left\\{\mu_{\beta}\bigl{(}L_{g,\varepsilon}^{+}(\mathrm{e}_{1},s_{1})\bigr{)},\mu_{\beta}\Bigl{(}L_{g,\varepsilon}^{+}\bigl{(}\mathrm{e}_{1},w_{1}(1/s_{2})\bigr{)}\Bigr{)}\right\\}\end{split}$

for all $0\leq s_{1}\leq b_{\varepsilon}$ and all
$0<1/s_{2}\leq\|h\|_{\infty}$, where again the notations
$L_{\varepsilon}^{+}(u,c)$ and $L_{g,\varepsilon}^{+}(u,c)$ represent the sets
$\bigl{(}L_{\varepsilon}\bigr{)}^{+}(u,c)$ and
$\bigl{(}L_{g,\varepsilon}\bigr{)}^{+}(u,c)$, respectively, for a given
direction $u\in\mathbb{S}^{n-1}$ and a real number $c\in\mathbb{R}$.

So, defining $w_{1}(s):=\gamma$ if $s\geq\|h\|_{\infty}$, and taking into
account that $b_{\varepsilon}\leq b\leq\delta$ (by (iii) in (3.3)), from the
fact that $\alpha<\beta$ and using Fubini’s theorem we have on the one hand
that

(3.15) $\begin{split}&\\!\frac{1}{\beta-\alpha}\int_{L_{\varepsilon}}\langle
x,\mathrm{e}_{1}\rangle\langle
x,\mathrm{e}_{2}\rangle^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)\\\
&\quad\,=\int_{L_{\varepsilon}}\int_{0}^{\langle
x,\mathrm{e}_{1}\rangle}\mathrm{d}s_{1}\int_{0}^{1/\langle
x,\mathrm{e}_{2}\rangle}s_{2}^{\beta-\alpha-1}\,\mathrm{d}s_{2}\,\mathrm{d}\mu_{\beta}(x)\\\
&\quad\,=\int_{0}^{b_{\varepsilon}}\int_{0}^{1/\varepsilon}s_{2}^{\beta-\alpha-1}\mu_{\beta}\bigl{(}L_{\varepsilon}^{+}(\mathrm{e}_{1},s_{1})\cap
L_{\varepsilon}^{-}(\mathrm{e}_{2},1/s_{2})\bigr{)}\,\mathrm{d}s_{2}\,\mathrm{d}s_{1}\\\
&\quad\,<\int_{0}^{b_{\varepsilon}}\int_{0}^{1/\varepsilon}s_{2}^{\beta-\alpha-1}\min\left\\{\mu_{\beta}\bigl{(}L_{\varepsilon}^{+}(\mathrm{e}_{1},s_{1})\bigr{)},\mu_{\beta}\bigl{(}L_{\varepsilon}^{-}(\mathrm{e}_{2},1/s_{2})\bigr{)}\right\\}\,\mathrm{d}s_{2}\,\mathrm{d}s_{1}\\\
&\quad\,\leq\int_{0}^{\delta}\int_{0}^{1/\varepsilon}s_{2}^{\beta-\alpha-1}\frac{b}{\beta}\varepsilon^{\beta}\,\mathrm{d}s_{2}\,\mathrm{d}s_{1}\\\
&\quad\,+\int_{0}^{\delta}\int_{0}^{1/\varepsilon}s_{2}^{\beta-\alpha-1}\min\left\\{\mu_{\beta}\bigl{(}L_{g,\varepsilon}^{+}(\mathrm{e}_{1},s_{1})\bigr{)},\mu_{\beta}\Bigl{(}L_{g,\varepsilon}^{+}\bigl{(}\mathrm{e}_{1},w_{1}(1/s_{2})\bigr{)}\Bigr{)}\right\\}\,\mathrm{d}s_{2}\,\mathrm{d}s_{1}\\\
&\quad\,=\frac{b\delta}{\beta-\alpha}\varepsilon^{\alpha}+\int_{0}^{\delta}\int_{0}^{1/\varepsilon}s_{2}^{\beta-\alpha-1}\mu_{\beta}\Bigl{(}L_{g,\varepsilon}^{+}(\mathrm{e}_{1},s_{1})\cap
L_{g,\varepsilon}^{+}\bigl{(}\mathrm{e}_{1},w_{1}(1/s_{2})\bigr{)}\Bigr{)}\,\mathrm{d}s_{2}\,\mathrm{d}s_{1}\\\
&\quad\,=\frac{b\delta}{\beta-\alpha}\varepsilon^{\alpha}+\frac{1}{\beta-\alpha}\int_{L_{g,\varepsilon}}\langle
x,\mathrm{e}_{1}\rangle\varphi_{1}(\langle
x,\mathrm{e}_{1}\rangle)^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x),\end{split}$

where in the last equality above we have also used that $\langle
x,\mathrm{e}_{1}\rangle\geq w_{1}(1/s_{2})$ if and only if
$1/\varphi_{1}(\langle x,\mathrm{e}_{1}\rangle)$ $\geq s_{2}$ (since
$\varphi_{1}$ is decreasing).

On the other hand, since $\alpha<\beta$, from Fubini’s theorem (jointly with
(ii) in (3.12)) we have that

(3.16) $\begin{split}\frac{1}{\beta-\alpha}\int_{L_{\varepsilon}}\langle
x,\mathrm{e}_{2}\rangle^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)&=\int_{L_{\varepsilon}}\int_{0}^{1/\langle
x,\mathrm{e}_{2}\rangle}s^{\beta-\alpha-1}\,\mathrm{d}s\,\mathrm{d}\mu_{\beta}(x)\\\
&=\int_{0}^{1/\varepsilon}s^{\beta-\alpha-1}\mu_{\beta}\bigl{(}L_{\varepsilon}^{-}(\mathrm{e}_{2},1/s)\bigr{)}\,\mathrm{d}s\\\
&=\int_{0}^{1/\varepsilon}s^{\beta-\alpha-1}\mu_{\beta}\Bigl{(}L_{g,\varepsilon}^{+}\bigl{(}\mathrm{e}_{1},w_{1}(1/s)\bigr{)}\Bigr{)}\,\mathrm{d}s\\\
&=\frac{1}{\beta-\alpha}\int_{L_{g,\varepsilon}}\varphi_{1}(\langle
x,\mathrm{e}_{1}\rangle)^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x),\end{split}$

where in the last equality above we have used again that $\langle
x,\mathrm{e}_{1}\rangle\geq w_{1}(1/s)$ if and only if $1/\varphi_{1}(\langle
x,\mathrm{e}_{1}\rangle)$ $\geq s$. Hence, from (3.15) and (3.16), we obtain
(3.13), as desired.

Now we will prove that for any concave function
$\varphi_{1}:[\gamma,\delta]\longrightarrow[0,\infty)$ we have that

(3.17) $\frac{\int_{L_{g,\varepsilon}}\langle
x,\mathrm{e}_{1}\rangle\varphi_{1}(\langle
x,\mathrm{e}_{1}\rangle)^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)}{\int_{L_{g,\varepsilon}}\varphi_{1}(\langle
x,\mathrm{e}_{1}\rangle)^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)}\leq\frac{\int_{L_{g,\varepsilon}}\langle
x,\mathrm{e}_{1}\rangle\left(\delta-\langle
x,\mathrm{e}_{1}\rangle\right)^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)}{\int_{L_{g,\varepsilon}}\left(\delta-\langle
x,\mathrm{e}_{1}\rangle\right)^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)}.$

To this aim, let $C_{1}>0$ be such that

(3.18) $\int_{L_{g,\varepsilon}}\bigl{(}C_{1}(\varepsilon)(\delta-\langle
x,\mathrm{e}_{1}\rangle)\bigr{)}^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)=\int_{L_{g,\varepsilon}}\varphi_{1}(\langle
x,\mathrm{e}_{1}\rangle)^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x).$

Since the latter identity is equivalent (by Fubini’s theorem) to

$\int_{\gamma}^{w_{1}(\varepsilon)}\left(\bigl{(}C_{1}(\varepsilon)(\delta-t)\bigr{)}^{\alpha-\beta}-\varphi_{1}(t)^{\alpha-\beta}\right)g^{\beta}(t)\,\mathrm{d}t=0,$

we may assert, taking into account that $\varphi_{1}$ is concave, that there
exists $t_{0}(\varepsilon)\in(\gamma,\delta)$ such that

(3.19)
$\begin{array}[]{l}\text{(i)}\;C_{1}(\varepsilon)(\delta-t)\geq\varphi_{1}(t)\text{
for every }\gamma\leq t\leq t_{0}(\varepsilon),\quad\text{and}\\\\[17.07164pt]
\text{(ii)}\;C_{1}(\varepsilon)(\delta-t)\leq\varphi_{1}(t)\text{ for every
}t_{0}(\varepsilon)\leq t\leq w_{1}(\varepsilon).\end{array}$

Then, from (3.18) and (3.19) (taking into account that $\alpha<\beta$), and
using Fubini’s theorem, we obtain

$\begin{split}\int_{L_{g,\varepsilon}}\langle
x,\mathrm{e}_{1}\rangle&\varphi_{1}(\langle
x,\mathrm{e}_{1}\rangle)^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)-\int_{L_{g,\varepsilon}}\langle
x,\mathrm{e}_{1}\rangle\bigl{(}C_{1}(\varepsilon)(\delta-\langle
x,\mathrm{e}_{1}\rangle)\bigr{)}^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)\\\
=&\,\frac{1}{\beta}\int_{\gamma}^{w_{1}(\varepsilon)}t\Bigl{(}\varphi_{1}(t)^{\alpha-\beta}-\bigl{(}C_{1}(\varepsilon)(\delta-t)\bigr{)}^{\alpha-\beta}\Bigr{)}g^{\beta}(t)\,\mathrm{d}t\\\
=&\,\frac{1}{\beta}\int_{\gamma}^{t_{0}(\varepsilon)}t\Bigl{(}\varphi_{1}(t)^{\alpha-\beta}-\bigl{(}C_{1}(\varepsilon)(\delta-t)\bigr{)}^{\alpha-\beta}\Bigr{)}g^{\beta}(t)\,\mathrm{d}t\\\
+&\,\frac{1}{\beta}\int_{t_{0}}^{w_{1}(\varepsilon)}t\Bigl{(}\varphi_{1}(t)^{\alpha-\beta}-\bigl{(}C_{1}(\varepsilon)(\delta-t)\bigr{)}^{\alpha-\beta}\Bigr{)}g^{\beta}(t)\,\mathrm{d}t\\\
\leq&\,\frac{t_{0}(\varepsilon)}{\beta}\int_{\gamma}^{t_{0}(\varepsilon)}\left(\varphi_{1}(t)^{\alpha-\beta}-\bigl{(}C_{1}(\varepsilon)(\delta-t)\bigr{)}^{\alpha-\beta}\right)g^{\beta}(t)\,\mathrm{d}t\\\
+&\,\frac{t_{0}(\varepsilon)}{\beta}\int_{t_{0}(\varepsilon)}^{w_{1}(\varepsilon)}\left(\varphi_{1}(t)^{\alpha-\beta}-\bigl{(}C_{1}(\varepsilon)(\delta-t)\bigr{)}^{\alpha-\beta}\right)g^{\beta}(t)\,\mathrm{d}t\\\
=&\,\frac{t_{0}(\varepsilon)}{\beta}\int_{\gamma}^{w_{1}(\varepsilon)}\left(\varphi_{1}(t)^{\alpha-\beta}-\bigl{(}C_{1}(\varepsilon)(\delta-t)\bigr{)}^{\alpha-\beta}\right)g^{\beta}(t)\,\mathrm{d}t\\\
=&\,t_{0}(\varepsilon)\int_{L_{g,\varepsilon}}\left(\varphi_{1}(\langle
x,\mathrm{e}_{1}\rangle)^{\alpha-\beta}-\bigl{(}C_{1}(\varepsilon)(\delta-\langle
x,\mathrm{e}_{1}\rangle)\bigr{)}^{\alpha-\beta}\right)\,\mathrm{d}\mu_{\beta}(x)=0.\end{split}$

Thus, we have

$\int_{L_{g,\varepsilon}}\langle x,\mathrm{e}_{1}\rangle\varphi_{1}(\langle
x,\mathrm{e}_{1}\rangle)^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)\leq\int_{L_{g,\varepsilon}}\langle
x,\mathrm{e}_{1}\rangle\bigl{(}C_{1}(\varepsilon)(\delta-\langle
x,\mathrm{e}_{1}\rangle)\bigr{)}^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x),$

which, together with (3.18), yields (3.17).

Hence, from (3.13) and (3.17), for every $0<\varepsilon\leq\|h\|_{\infty}$ we
get

$\begin{split}\frac{\int_{L_{\varepsilon}}\langle
x,\mathrm{e}_{1}\rangle\langle
x,\mathrm{e}_{2}\rangle^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)}{\int_{L_{\varepsilon}}\langle
x,\mathrm{e}_{2}\rangle^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)}&<\frac{\int_{L_{g,\varepsilon}}\langle
x,\mathrm{e}_{1}\rangle\left(\delta-\langle
x,\mathrm{e}_{1}\rangle\right)^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)}{\int_{L_{g,\varepsilon}}\left(\delta-\langle
x,\mathrm{e}_{1}\rangle\right)^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)}\\\\[5.69054pt]
&+\frac{b\delta\varepsilon^{\alpha}}{\int_{L_{\varepsilon}}\langle
x,\mathrm{e}_{2}\rangle^{\alpha}\,\mathrm{d}x}.\end{split}$

Now, taking limits as $\varepsilon\to 0^{+}$ in the above inequality, we have
that the left-hand side, namely,

$\frac{\int_{L_{\varepsilon}}\langle x,\mathrm{e}_{1}\rangle\langle
x,\mathrm{e}_{2}\rangle^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)}{\int_{L_{\varepsilon}}\langle
x,\mathrm{e}_{2}\rangle^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)}=\frac{\int_{L_{\varepsilon}}\langle
x,\mathrm{e}_{1}\rangle\langle
x,\mathrm{e}_{2}\rangle^{\alpha-1}\,\mathrm{d}x}{\int_{L_{\varepsilon}}\langle
x,\mathrm{e}_{2}\rangle^{\alpha-1}\,\mathrm{d}x},$

tends to

$\frac{\int_{L}\langle x,\mathrm{e}_{1}\rangle\langle
x,\mathrm{e}_{2}\rangle^{\alpha-1}\,\mathrm{d}x}{\int_{L}\langle
x,\mathrm{e}_{2}\rangle^{\alpha-1}\,\mathrm{d}x}=\mathrm{g}_{\alpha}(h)$

as $\varepsilon\to 0^{+}$ (see (3.1)). Furthermore, the first term in the
right-hand side,

$\frac{\int_{L_{g,\varepsilon}}\langle
x,\mathrm{e}_{1}\rangle\left(\delta-\langle
x,\mathrm{e}_{1}\rangle\right)^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)}{\int_{L_{g,\varepsilon}}\left(\delta-\langle
x,\mathrm{e}_{1}\rangle\right)^{\alpha-\beta}\,\mathrm{d}\mu_{\beta}(x)}=\frac{\frac{c^{\beta}}{\beta}\int_{\gamma}^{w_{1}(\varepsilon)}t(\delta-t)^{\alpha}\,\mathrm{d}t}{\frac{c^{\beta}}{\beta}\int_{\gamma}^{w_{1}(\varepsilon)}(\delta-t)^{\alpha}\,\mathrm{d}t},$

tends to

$\mathrm{g}_{\alpha}(g)=\frac{\int_{\gamma}^{\delta}t(\delta-t)^{\alpha}\,\mathrm{d}t}{\int_{\gamma}^{\delta}(\delta-t)^{\alpha}dt}=\left(\delta-\frac{(\alpha+1)(\delta-\gamma)}{\alpha+2}\right)$

for $\varepsilon\to 0^{+}$, whereas the second term in the right-hand side,

$\frac{b\delta\varepsilon^{\alpha}}{\int_{L_{\varepsilon}}\langle
x,\mathrm{e}_{2}\rangle^{\alpha}\,\mathrm{d}x},$

clearly tends to

$\frac{0}{\int_{L}\langle x,\mathrm{e}_{2}\rangle^{\alpha}\,\mathrm{d}x}=0$

for $\varepsilon\to 0^{+}$. Therefore, we have that

$\mathrm{g}_{\alpha}(h)\leq\mathrm{g}_{\alpha}(g).$

Finally, the latter relation jointly with (ii) and (iii) in (3.2) gives us

$\begin{split}\frac{\int_{\mathrm{g}_{\alpha}(h)}^{b}h(t)^{\beta}\,\mathrm{d}t}{\int_{0}^{b}h(t)^{\beta}\,\mathrm{d}t}&=\frac{\int_{\mathrm{g}_{\alpha}(h)}^{\delta}g(t)^{\beta}\,\mathrm{d}t}{\int_{\gamma}^{\delta}g(t)^{\beta}\,\mathrm{d}t}\geq\frac{\int_{\mathrm{g}_{\alpha}(g)}^{\delta}g(t)^{\beta}\,\mathrm{d}t}{\int_{\gamma}^{\delta}g(t)^{\beta}\,\mathrm{d}t}=\left(\frac{\delta-\mathrm{g}_{\alpha}(g)}{\delta-\gamma}\right)^{\beta+1}\\\
&=\left(\frac{\alpha+1}{\alpha+2}\right)^{\beta+1},\end{split}$

as desired. This finishes the proof of (1.8), and hence also that of Theorem
1.2.

## References

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|  Self-Folding and Self-Scrolling Mechanisms of Edge-Deformed Graphene
Sheets: A Molecular Dynamics Study
---|---
|  Marcelo Lopes Pereira Juniora, and Luiz Antonio Ribeiro Juniora,∗
|  Graphene-based nanofolds (GNFs) are edge-connected 2D stacked monolayers
originated from single-layer graphene. Graphene-based nanoscrolls (GNSs) are
nanomaterials with geometry resembling graphene layers rolled up into a spiral
(papyrus-like) form. Both GNSs and GNFs structures induce significant changes
in the mechanical and optoelectronic properties of single-layer graphene,
aggregating new functionalities in carbon-based applications. Here, we carried
our fully atomistic reactive (ReaxFF) molecular dynamics simulations to study
the self-folding and self-scrolling of edge-deformed graphene sheets. We
adopted initial armchair edge-scrolled graphene (AESG($\phi$,$\theta$))
structures with similar (or different) twist angles ($\phi$,$\theta$) in each
edge, mimicking the initial configuration that was experimentally developed to
form biscrolled sheets. Results showed that AESG(0,$2\pi$) and
AESG(2$\pi$,$2\pi$) evolved to single-folded and two-folded fully stacked
morphologies, respectively. As a general trend, for twist angles higher than
$2\pi$, the self-deformation process of AESG morphologies yields GNSs. Edge
twist angles lower than $\pi$ are not enough for triggering the self-
deformation processes. In the AESG(0,3$\pi$) and AESG(3$\pi$,3$\pi$) cases,
after a relaxation period, their morphology transition towards GNS occurred
rapidly. In the AESG(3$\pi$,3$\pi$) dynamics, a metastable biscroll was formed
by the interplay between the left- and right-sided partial scrolling in
forming a unique GNS. At high-temperature perturbations, the edge folding and
scrolling transitions to GNFs and GNSs occurred within the ultrafast period.
Remarkably, the AESG(2$\pi$,3$\pi$) evolved to a dual state that combines
folded and scrolled structures in a temperature-independent process.

##

††footnotetext: a Institute of Physics, University of Brasília, Brasília
70919-970, Brazil.††footnotetext: Corresponding Author<EMAIL_ADDRESS>(L.A.R.J.)††footnotetext: † Electronic Supplementary Information (ESI)
available: The Supplementary Material presents the videos for MD simulations
of all the systems studied here.

##  1 Introduction

Fig. 1: Schematic representation of the model AESG lattices studied here.(a)
zooms-in a region of the adopted graphene sheet, which is presented in (b).
(c) and (d) illustrate, respectively, the one-edge-scrolled graphene
(AESG(0,$\theta$)) and two-edge-scrolled graphene (AESG($\phi$,$\theta$))
models used as input structures. (e) shows the edge structure configuration
with its inner ($r_{1}$) and outer ($r_{2}$) radius. $\phi$ can assume the
following values: $-3\pi$, $-2\pi$, 0, $2\pi$, and $3\pi$. $\theta$ can assume
the following values: 0, $2\pi$, and $3\pi$.

Graphene has been evolved into a field of unprecedented activity in physics,
chemistry, and materials research 1. Several of its unique properties
originate from its two-dimensional layer of one-atom thickness 2. Due to its
excellent in-plane mechanical flexibility and strong inter-layer van der Waals
interactions, single-layer graphene (SLG) is susceptible to deforming into
scrolled 3 and folded 4 configurations. Graphene nanoscrolls (GNSs) are
composed of tubularly scrolled SLG, yielding a 1D topological structure 5, 6,
7, 8. Some GNSs capabilities, such as superlubricity and large current
sustainability, surpassing the properties of SLG itself 9. Graphene nanofolds
(GNFs), in turn, are edge-connected 2D stacked monolayers that also originated
from SLG. Their continuous and curved edge — with a twist angle ($\theta$)
between the two membranes — can exhibit different band structures
$\theta$-dependent interfacial conductivity 10. Both GNSs and GNFs structures
induce significant changes in the mechanical and optoelectronic properties of
SLG, aggregating new functionalities in graphene-based applications 11, 12,
13, 14, 15, 16.

Several experimental and theoretical studies have been performed to elucidate
the self-scrolling 13, 5, 17, 18, 19, 20 and self-folding 21, 10, 22, 23, 24,
25 processes of SLG. Experimentally, high-resolution atomic force microscopy
(AFM) 21, 26, scanning electron microscopy (SEM) 27, 28, and transmission
electron microscopy (TEM) 4, 13 have been employed in the study of SLG folding
or scrolling deformations on flat substrates or in suspended conditions. In a
theoretical perspective, continuum mechanics (CM) and molecular dynamics (MD)
simulation techniques were employed to also investigate the SLG folding and
scrolling behaviors 5, 29, 30, 31, 32, 12. Importantly, these theoretical
studies provide useful insights on the GNS and GNF formations, some of which
may not be readily obtainable in experimental measurements (e.g., graphene
local folding curvatures or the potential energy landscape for the non-bonded
interaction terms). As a general trend, these works concluded that the self-
formation of both GNSs and GNFs results from a interplay between the SLG
bending rigidity and the van der Waals interactions.

Different analytical models 33, 34 and experimental setups 11 have been
proposed to study the scrolling and folding mechanics of SLG. Recently, Li et.
al proposed a finite-deformation analytical model in which no presumed
assumptions on the geometries of deformed configurations are required 33. Both
scroll-predicted and fold-predicted profiles for SLG and their critical
conditions showed good agreement with MD simulation results. Lima et. al
developed an experimental approach that involves a twist-based spinning of
carbon-based layers to form a biscrolled sheet 11. In that case, they
converted a carbon nanotube sheet into a biscrolled nanotube sheet. The
initial setup for the experiment was based on a structure that presents a
cross-section similar to the illustration in Figure 1(d). Their findings
revealed that the ability to make biscrolled yarns can be exploited to
optimize yarn properties. Despite the amount of successful studies on the SLG
conversion into GNSs and GNFs, their formation from a edge-deformed SLG is not
fully understood.

The self-folding 35 and self-scrolling 36 mechanisms of edge-deformed graphene
sheets were experimentally reported recently. Shin and coworkers 35 introduced
an ice-templated self-assembly approach for integrating two-dimensional
graphene nanosheets to fabricate graphene nanoscroll networks. They showed
that freeze-casting of reduced graphene oxide solution results in the
formation of graphene nanoscroll networks at pH 10. In addition, they have
demonstrated that graphene nanoscroll networks show good electrocatalytic
activity for the oxygen reduction reaction. In the synthesis of graphene
nanoscrolls, the self-scrolling process started from an edge-curved graphene
sheet that triggers the self-scrolling process of graphene, producing
nanoscrolls stabilized by the $\pi$-$\pi$ interactions. Meyer et al. 36
observed folded edges in the fabrication of monolayer and bilayer graphene.
During their fabrication processes, one of the edges was curved, forming
graphene folds spontaneously. Importantly, these investigations do not provide
information on the time needed for accomplishing the self-scrolling or self-
folding of edge-deformed graphene and for the possible products yielded in
these processes. In this sense, our investigation can shed light on the self-
deformation of curved graphene sheets contributing to their better
understanding.

In the present work, motivated by the deformation mechanism of carbon-based
sheets proposed by Lima et al 11, the self-folding and self-scrolling of a
edge-deformed SLG was investigated in the framework of fully-atomistic
reactive molecular dynamics simulations. Our computational approach was based
on adopting initial armchair edge-scrolled graphene (AESG) structures with
similar (or different) twist angles in each edge, mimicking the initial
configuration used to form biscrolled sheets reported in reference 11.
Moreover, we also provided information on the folding and scrolling curvatures
and the potential energy landscapes during the AESG deformation processes.

##  2 Methodology

To study the self-folding and self-scrolling of AESG sheets, we carried out
fully-atomistic MD simulations within the scope of the reactive force field
ReaxFF potential 37, 38, as implemented in the Large-scale Atomic/Molecular
Massively Parallel Simulator (LAMMPS) 39. Specifically, we used the parameters
set provided by Chenoweth et al. for C/H/O 40.

The initial configurations of our model AESG systems are depicted in Figure 1.
Figure 1(a) zooms-in a region of the graphene sheet used here, which is
presented in Figure 1(b). Figures 1(c) and 1(d) illustrate, respectively, the
one-edge-scrolled graphene (AESG(0,$\theta$)) and two-edge-scrolled graphene
(AESG($\phi$,$\theta$)) models used in our simulations as input structures.
$\theta$ and $\phi$ allow different twist angles on each edge. Figure 1(e)
shows the edge structure configuration with its inner ($r_{1}$) and outer
($r_{2}$) radius. The graphene sheet has $x$ and $y$ dimensions of 100 Å and
200 Å, respectively, totaling 7776 atoms. These structures were atomistically
modeled using the Sculptor plug-in 41, as implemented in the software Visual
Molecular Dynamics (VMD) 41. VMD was also used here to obtain the MD snapshots
and trajectories.

The equations of motion were numerically integrated using the velocity-Verlet
integrator with a time-step of $0.25$ fs until 250 ps. We adopted an NVT
ensemble and Nosé-Hoover thermostat 42, with a time constant of 100 fs, to
analyze the influence of temperature on the scrolling and folding behaviors.
The temperature regimes ranged from 100 K up to 1000 K with an increment of
100 K. Initial atomic velocities were randomly generated for each temperature.

Fig. 2: Representative MD snapshots for the self-deformation process of (a-e)
AESG(0,2$\pi$) and (f-j) AESG(2$\pi$,2$\pi$) at 300 K. Importantly, the panels
in this figure illustrate the slide motion obtained during the fold formation.
In panels (c) and (d), the graphene edge on top moves towards the left
direction, whereas the graphene edge at the bottom moves towards the right
one. This trend of motion is better understood by watching the supplementary
video SM1. Fig. 3: Representative MD snapshots for the self-deformation
process of (a-e) AESG(0,3$\pi$) and (f-j) AESG(3$\pi$,3$\pi$) at 300 K.

##  3 Results

Fig. 4: Representative MD snapshots for the self-deformation process of
AESG($2\pi$,3$\pi$) at 300 K.

We begin our discussion by presenting the MD snapshots for the folding and
scrolling mechanisms of the model AESG systems. The Supplementary Material
presents the videos for MD simulations of all the systems studied here at 300
K and 1000 K. Figures 2(a-e) (and supplementary video SM1) and 2(f-j) (and
supplementary video SM2) illustrate the representative MD snapshots for the
self-deformation process of AESG(0,2$\pi$) and AESG(2$\pi$,2$\pi$) cases at
300 K, respectively. In Figures 2(a) and 2(b) one can note that a smooth
morphology transition, from a twisted edge towards a folded one, occurs within
an ultrafast time-period, about 2 ps. It is the period necessary for the
system to escape from the local potential well at this specific thermal
excitation. For higher temperature perturbations, the edge folding and
scrolling transitions occur much faster. At 1000 K, the transition mechanism
illustrated in Figures 2(a-e) also takes place, but before 1 ps (see
supplementary video SM3). During 240 ps, the staked monolayers formed in the
morphology transition slide against each other, as shown in Figures 2(c) and
2(d). After this transient period, a fully GNF is formed at 250 ps (Figure
2(e)). It is worth noting that this fully single-folded structure was also
obtained at 1000 K (see supplementary video SM3). For the AESG(2$\pi$,2$\pi$)
case (Figures 2(f-j)), the edge scrolling-folding transition occurs in the
same fashion, but a two-folded structure was yielded instead. This two-folded
structure is only formed below a critical temperature regime, about 900 K.
Above this temperature threshold, the random motion of the atoms, imposed by
the thermal fluctuations, overcomes the potential energy well (bond energies
plus van der Waals interactions) related to the formation of the two-folded
structure, and it cannot overcome the potential energy well of a single-folded
one. In this way, the edge scrolling-folding transition at 1000 K leads to a
fully single-folded morphology (see supplementary video SM4).

Figures 3(a-e) (and supplementary video SM5) and 3(f-j) (and supplementary
video SM6) illustrate the representative MD snapshots for the self-deformation
process of AESG(0,3$\pi$) and AESG(3$\pi$,3$\pi$) cases at 300 K,
respectively. As a general trend, for twist angles higher than $2\pi$, the
self-deformation process of AESG yields GNSs morphologies. Edge twist angles
lower than $\pi$ are not enough for triggering the self-deformation processes
once the interlayer van der Waals interactions in the edge can not overcome
the SLG bending rigidity. Consequently, the AESG structure is converted into a
conventional SLG. As shown in Figures 3(a) and 3(b), the AESG(0,3$\pi$) leads
about 2 ps to relax into a form that roughly preserves the initial size of the
scrolled part. After this relaxation period, the transition from
AESG(0,3$\pi$) towards a GNS morphology occurs rapidly (about 30 ps),
according to Figures 3(c) and 3(d). At about 50 ps, a stable GNS with entirely
overlapped layers is formed and keeps its integrity until the end of the
simulation (Figure 3(e)). As expected, Figures 3(f) and 3(g) show that the
self-deformation process of AESG(3$\pi$,3$\pi$) presents a relaxation period
similar to the AESG(3$\pi$,3$\pi$) case. At 15 ps, a metastable biscroll is
formed by the interplay between the left- and right-sided partial scrolling in
forming a unique GNS, as illustrated in Figure 3(h). At 75 ps, the right side
wins and a single GNS starts to be formed (see Figure 3(i)) and a GNS with
entirely overlapped layers is formed at 76 ps, remaining stable until the
simulation, as depicted in Figure 3(j).

Note that there is no preferred side in converting a metastable biscroll into
a GNS. This is a random process that depends on the seed used to randomly
generate the atom velocities in LAMMPS. Moreover, the thermal fluctuations
imposed on the displacement of the atoms are randomly obtained and can also
impact the conversion process of a metastable biscroll into a GNS. Differently
to what was obtained for the folding cases (Figure 2), the AESG(0,3$\pi$) and
AESG(3$\pi$,3$\pi$) MD formed stable GNSs even at high temperatures. See
supplementary videos SM7 and SM8 for the AESG(0,3$\pi$) and
AESG(3$\pi$,3$\pi$) MD at 1000 K, respectively. In the supplementary videos
SM5-SM8, one can note that the GNS presents a breathing motion that varies the
inner radius amplitude in an oscillatory fashion. This behavior was detailed
discussed in references 43.

We now present in Figure 4 (and in the supplementary video SM9) the
representative MD snapshots for the AESG(2$\pi$,3$\pi$) case, in which the
edges were initially scrolled with different twist angles. Remarkably, the
AESG(2$\pi$,3$\pi$) evolves to a dual state that combines folded and scrolled
structures. This process is temperature-independent (see supplementary videos
SM9 and SM10 (AESG(2$\pi$,3$\pi$) at 1000 K)). The AESG(2$\pi$,3$\pi$)
relaxation process illustrated in Figures 4(a) and 4(b) occurs according to
the mechanisms described above for the AESG(0,2$\pi$) and AESG(0,3$\pi$) cases
(Figures 2) and 3, respectively). In Figure 4(c), one can realize that edge
initially scrolled with $2\pi$ tends to form a GNF whereas the edge scrolled
with $3\pi$ tries to form a GNS. At 75 ps, the partially formed fold and
scroll overlap preserving their forms (see Figure 4(d)). The final structure
obtained in the AESG(2$\pi$,3$\pi$) MD is presented in Figure 4(d). It is
worth mentioning that the coexistence of folded and scrolled shapes of
graphene in the final morphology presented Figure 4(d) has not been reported
so far.

Fig. 5: Time evolution of the potential energy for MD simulations presented
above: (a) AESG(0,2$\pi$), (b) AESG(2$\pi$,2$\pi$), (c) AESG(2$\pi$,3$\pi$),
(d) AESG(0,3$\pi$), and (e) AESG(3$\pi$,3$\pi$).

In the context of our simulations, the edge topology does not affect the
folding and scrolling mechanisms of the graphene sheets. The Supplementary
Material videos SM11-SM15 show the dynamics of zigzag ESG (ZESG) at 300 K,
considering all the initial structures modeled here. The supplementary videos
SM11, SM12, SM13, SM14, and SM15 are referring to the ZESG(0,2$\pi$),
ZESG(2$\pi$,2$\pi$), ZESG(2$\pi$,3$\pi$), ZESG(0,3$\pi$), and
ZESG(3$\pi$,3$\pi$) cases, respectively. By contrasting the results presented
in the supplementary videos for the armchair cases at 300 K (SM1, SM3, SM5,
SM7, and SM9) with the ones for the zigzag lattices (SM11-SM15), one can note
that the zigzag and armchair ESG dynamics are similar (i.e., not edge-
sensitive).

We carried out simulations considering negative values of $\phi$. These
simulations are presented in the Supplementary Material (videos SM16-SM18)
alongside their related plots for the time evolution of the potential energy.
The supplementary videos SM16, SM17, and SM18 are referring to the
AESG(-2$\pi$,2$\pi$), AESG(-3$\pi$,2$\pi$), and AESG(-3$\pi$,3$\pi$)
cases,respectively. For the AESG(-2$\pi$,2$\pi$) case, the system evolves to a
collapsed structure with two-folded edges due to the formation of carbon-
carbon covalent bonds since the edges are very close to each other. The
AESG(-2$pi$,3$\pi$) case evolves to a single CNS, and the AESG(-3$\pi$,3$\pi$)
case yields a dual state that combines folded and scrolled structures, similar
to the results presented in Figure 4(e). Figure S3 in Supplementary Material
illustrates the representative MD snapshots for the AESG(-2$\pi$,2$\pi$) case,
to highlight the collapsed structure with two-folded edges.

Finally, Figure 5 shows the time evolution of the potential energy for all the
cases discussed above. In this figure, the structural stability of formed
folded, scrolled, and dual-species is characterized by the convergence of the
curves. For the sake of comparison, the blue, red, and green dashed lines
denote the value for the average potential energies (in the last 100 ps) for
the AESG(0,2$\pi$), AESG(0,3$\pi$), and AESG(2$\pi$,3$\pi$), respectively.
Note that this average value is similar among the related AESG(0,$\theta$) and
AESG($\theta$,$\theta$) cases. As expected, the curve profiles in Figure 5
suggest that scrolled and folded configurations are energetically favorable.
As observed in Figures 5(a) and 5(b), the potential energy oscillates in time
due to the relative slide motion between the folded layers. This motion
increases (or decreases) the van der Waals interactions depending on the
stacked area in the GNF formation. The potential energy related to the folding
processes is slightly higher than the stabilization energy in the scrolling
processes (red dashed line) and slightly lower than the one obtained in
forming the dual-species (green dashed line). In Figure 5(c), one can see that
the potential energy in forming the dual state that combines folded and
scrolled structures converges rapidly. Since the converged potential energy is
marginally higher than the one for AESG(0,$\theta$) and
AESG($\theta$,$\theta$) cases, the dual state is also energetically favorable.
Figures 5(d) and 5(e) show that, during the dynamical process for the GNSs
formation, there is a downhill trend for the potential energy values that ends
about 40 ps and 100 ps for AESG(0,3$\theta$) and AESG(3$\theta$,3$\theta$),
respectively. From that moment on, the total potential energy stabilizes and a
flat region persists until the end of the simulation. As mentioned above, the
collapsed CNSs are formed within the ultrafast time scale. The transient
plateau (20-100 ps) in Figure 5(e) refers to the formation of the metastable
biscroll (see Figure 3(h)). At high temperatures, GNFs and GNSs formation
energies are greater when compared, for instance, with results in Figure 5.

##  4 Conclusions

In summary, we carried out fully atomistic (ReaxFF) MD simulations to study
the self-folding and self-scrolling of a edge-deformed graphene sheet. We
adopted initial edge-scrolled graphene structures with similar (or different)
twist angles in each edge, mimicking the initial configuration used to form
biscrolled sheets reported in reference 11. Results showed that AESG(0,$2\pi$)
and AESG(2$\pi$,$2\pi$) evolved to single-folded and two-folded fully stacked
morphologies, respectively. The two-folded structure is only formed below a
critical temperature regime, about 900 K. The edge scrolling-folding
transition of AESG(2$\pi$,$2\pi$), at 1000 K, leads to a fully single-folded
morphology.

As a general trend, for twist angles higher than $2\pi$, the self-deformation
process of AESG morphologies yields GNSs. Edge twist angles lower than $\pi$
are not enough for triggering the self-deformation processes once the
interlayer van der Waals interactions in the edge can not overcome the SLG
bending rigidity. Consequently, the AESG structure is converted into a
conventional graphene monolayer. In the AESG(0,3$\pi$) and AESG(3$\pi$,3$\pi$)
cases, after a relaxation period, their morphology transitions towards GNS
occurred rapidly. In the AESG(3$\pi$,3$\pi$) MD, a metastable biscroll was
formed by the interplay between the left- and right-sided partial scrolling in
forming a unique GNS. Differently to what was obtained for the folding cases
(Figure 2), the AESG(0,3$\pi$) and AESG(3$\pi$,3$\pi$) MD formed stable GNSs
even at high temperatures. At high temperature perturbations, the edge folding
and scrolling transitions to GNFs and GNSs occurred within an ultrafast
period. Remarkably, the AAESG(2$\pi$,3$\pi$) evolves to a dual state that
combines folded and scrolled structures. This process is temperature-
independent. It is worth mentioning that the coexistence of folded and
scrolled shapes of graphene in the final morphology presented Figure 4(d) has
not been reported so far.

It is worth mentioning that some theoretical works in the literature carried
out MD simulations using carbon nanotubes 44, 45, 46, 47, graphene nanoribbons
48, diamond clusters 13, and other kinds of agents 49, 19 for triggering the
folding or scrolling processes of graphene. These works have not investigated
the folding and scrolling mechanism triggered by edge deformations. Relevant
experimental studies in literature reported self-folding and self-scrolling of
graphene originated by its edge deformations 35, 36. In this sense, our
results shed light on this crucial mechanism. To the best of our knowledge,
the conversion mechanism of edge-deformed graphene into folds and scrolls and
the possible products yielded from this conversion — such as the collapsed
structure with two-folded edges and the dual state that combines folded and
scrolled structures — are first reported here.

## Conflicts of interest

There are no conflicts to declare.

## Acknowledgements

The authors gratefully acknowledge the financial support from Brazilian
research agencies CAPES, CNPq, and FAP-DF. M.L.P.J. gratefully acknowledges
the financial support from CAPES grant 88882.383674/2019-01. L.A.R.J.
gratefully acknowledges the financial support from CNPq grant 302236/2018-0,
FAP-DF grant $0193.0000248/2019-32$, DPI/DIRPE/UnB (Edital DPI/DPG $03/2020$)
grant $23106.057541/2020-89$, and IFD/UnB (Edital $01/2020$) grant
$23106.090790/2020-86$. The authors gratefully acknowledge the National
Laboratory for Scientific Computing (LNCC/MCTI, Brazil) for providing HPC
resources of the SDumont supercomputer, which have contributed to the research
results reported within this work (http://sdumont.lncc.br). This research was
developed with the support of CENAPAD-SP (Centro Nacional de Processamento de
Alto Desempenho em São Paulo), grant UNICAMP / FINEP - MCT.

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