Some remarks on Petty projection of log-concave functions

TL;DR

This paper explores the Petty projection of log-concave functions, introduces new inequalities related to this concept, and corrects some previous results in the literature.

Contribution

It provides new inequalities for the Petty projection of log-concave functions and offers corrections to earlier findings in the field.

Findings

New inequalities involving Petty projection of log-concave functions

Corrections and complements to previous results in the literature

Enhanced understanding of the geometric properties of log-concave functions

Abstract

In this note we study the Petty projection of a log-concave function, which has been recently introduced in [9]. Moreover, we present some new inequalities involving this new notion, partly complementing and correcting some results from [9].

Equations

$$K^{\circ }=\{x\in {\mathbb{R}}^n:⟨x,y⟩\le 1\text{ for any }y\in K\},$$

$$h_K(x)=\sup \{⟨x,y⟩:y\in K\}.$$

$$V_1(K,L)=\frac{1}{n}\underset{\epsilon \to 0^{+}}{\lim }\frac{\mathrm{vol}(K+\epsilon L)-\mathrm{vol}(K)}{\epsilon }.$$

$$V_1(K,L)=\frac{1}{n}{\int }_{{\mathbb{S}}^{n-1}}{h}_L(v)dS(K,v),$$

$$h_{\Pi K}(u)={\mathrm{vol}}_{n-1}(P_{u^{\perp }}K),$$

$$h_{\Pi K}(u)=\frac{1}{2}{\int }_{{\mathbb{S}}^{n-1}}|⟨u,v⟩|dS(K,v).$$

$${\int }_{{\mathbb{S}}^{n-1}}|⟨u,v⟩|dS(K,v)={\int }_{{\mathbb{S}}^{n-1}}{h}_{L_u}(v)dS(K,v),$$

$$\mathrm{vol}(K+\epsilon L_u)-\mathrm{vol}(K)=2\epsilon {\int }_{P_{u^{\perp }}K}d{x}^{\prime }=2\epsilon {\mathrm{vol}}_{n-1}(P_{u^{\perp }}K).$$

$$n{V}_1(K,L_u)=\underset{\epsilon \to 0^{+}}{\lim }\frac{\mathrm{vol}(K+\epsilon L_u)-\mathrm{vol}(K)}{\epsilon }=2{\mathrm{vol}}_{n-1}(P_{u^{\perp }}K)=2h_{\Pi K}(u).$$

$$f((1-\lambda )x+\lambda y)\ge f(x{)}^{1-\lambda }f(y{)}^{\lambda },$$

$$I^{\infty}_{K}(x)=\left\{\begin{array}[]{lr}0&\text{if }\quad x\in K\\ \infty&\text{otherwise.}\end{array}\right.\quad\text{and}\quad\|x\|_{K}=\inf\{t>0:x\in tK\}.$$

$$h_K(x)=\sup \{⟨x,y⟩-I_K^{\infty }(y):y\in {\mathbb{R}}^n\}=(I_K^{\infty }{)}^{\ast }(x),$$

$${\phi }^{\ast }(x)=\sup \{⟨x,y⟩-\phi (y):y\in {\mathbb{R}}^n\}$$

$${\chi }_K=e^{-I_K^{\infty }}\text{ and }{h}_K=(I_K^{\infty }{)}^{\ast },$$

$$h_f={\phi }^{\ast },$$

$${\int }_{{\mathbb{S}}^{n-1}}|⟨u,v⟩|dS(K,v)={\int }_{{\mathbb{R}}^n}|⟨\nabla {\chi }_K(x),u⟩|dx,$$

$$h_{\Pi f}(u)=\frac{1}{2}{\int }_{{\mathbb{R}}^n}|⟨\nabla f(x),u⟩|dx,$$

$$h_{\Pi f}(u)=\frac{1}{2}{\int }_{\mathrm{supp}f}|⟨\nabla \phi (x),u⟩|f(x)dx.$$

$${\left({\int }_{{\mathbb{R}}^n}|\nabla f(x)|dx\right)}^n{\int }_{{\mathbb{R}}^n}{\Pi }^{\circ }f(z)dz\ge {\omega }_{n}n!{\left(\frac{n{\omega }_n}{{\omega }_{n-1}}\right)}^n.$$

$${\int }_{{\mathbb{R}}^n}|\nabla f(x)|dx\le n{\int }_{{\mathbb{R}}^n}f(y)dy+{\int }_{{\mathbb{R}}^n}f(z)\log \frac{f(z)}{a\Vert f{\Vert }_{\infty }}dz,$$

$$\frac{\left(\genfrac{}{}{0ex}{}{2n}{n}\right)}{n^n}\le \mathrm{vol}(K{)}^{n-1}\mathrm{vol}({\Pi }^{\circ }K)\le {\left(\frac{{\omega }_n}{{\omega }_{n-1}}\right)}^n.$$

$$h_{{\Pi }_{b}f}(y)={\int }_{suppf}|⟨\nabla \phi (x),y⟩|f(x)dx,\text{i.e.}{h}_{{\Pi }_{b}f}=2h_{\Pi f}.$$

$$\Vert x{\Vert }_{{\Pi }_b^{\circ }f}={\int }_{{\mathbb{R}}^n}|⟨\nabla f(x),y⟩|dx.$$

$$\frac{2^{-n}}{n!}\Vert f{\Vert }_1^{-n-1}{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}\min \{f(x),f(y)\}dxdy\le \mathrm{vol}({\Pi }_b^{\circ }f)\le {\left(\frac{{\omega }_n}{2{\omega }_{n-1}}\right)}^n\Vert f{\Vert }_{\frac{n}{n-1}}^{-n}.$$

$${\Pi }^{\circ }f(y)=e^{-h_{\Pi f}(y)}=e^{-\frac{1}{2}{h}_{{\Pi }_{b}f}(y)}=e^{-\frac{1}{2}\Vert y{\Vert }_{{\Pi }_b^{\circ }f}},$$

$${\int }_{{\mathbb{R}}^n}{\Pi }^{\circ }f(x)dx=2^{n}n!\mathrm{vol}({\Pi }_b^{\circ }f).$$

$$f_{\varepsilon}(x)=\left\{\begin{array}[]{lr}0&\text{if }d(x,K)\geq\varepsilon\\ 1-\frac{d(x,K)}{\varepsilon}&\text{if }d(x,K)<\varepsilon\end{array}\right.,$$

$$\nabla f_{\varepsilon}(x)=\left\{\begin{array}[]{lr}-\varepsilon^{-1}\nu(x^{\prime})&\text{if }x\in D_{\varepsilon}\\ 0&\text{otherwise}\\ \end{array}\right.,$$

$$\frac{1}{2}{\int }_{{\mathbb{R}}^n}|⟨\nabla f_{\epsilon }(x),y⟩|dx=\frac{1}{2}{\int }_{D_{\epsilon }}|⟨{\epsilon }^{-1}\nu (x^{\prime }),y⟩|dx=\frac{{\epsilon }^{-1}}{2}{\int }_{D_{\epsilon }}|⟨\nu (x^{\prime }),y⟩|dx.$$

$$\frac{{\epsilon }^{-1}}{2}{\int }_{D_{\epsilon }}|⟨\nu (x^{\prime }),y⟩|dx\to \frac{1}{2}{\int }_{\partial K}|⟨\nu (x^{\prime }),y⟩|d\sigma (\partial K,x^{\prime }),$$

Full text

1 Introduction

Let $K\subset\mathbb{R}^{n}$ be a convex body, i.e., a convex and compact set with non-empty interior, whose boundary is denoted by $\partial K$. Moreover, let $\mathcal{K}^{n}$ be the set of all convex bodies in $\mathbb{R}^{n}$. For these and most of the forthcoming definitions and ideas on Convex Geometry, we recommend the books [14] and [3].

If the origin is an interior point of $K$, the polar body $K^{\circ}$ of $K$ is

[TABLE]

which is also a convex body with the origin in its interior.

A convex body $K\in\mathcal{K}^{n}$ is uniquely defined by its support function, defined by

[TABLE]

For $C\subset\mathbb{R}^{n}$ of affine dimension $k\in\{1,\dots,n\}$, we denote by $\mathrm{vol}_{k}(C)$ its volume measured inside the affine hull of $C$, and moreover we also write $\mathrm{vol}(C)=\mathrm{vol}_{n}(C)$.

The mixed volume of two convex bodies $K$ ($n-1$ times) and $L$ can be defined by

[TABLE]

There is a unique finite measure $S(K,\cdot)$ on the unit euclidean sphere $\mathbb{S}^{n-1}$, called the surface area of $K$, so that

[TABLE]

(cf. [11]). When $K$ has a $\mathcal{C}^{2}$ boundary $\partial K$ with positive curvature, the density of $S(K,\cdot)$ with respect to the Lebesgue measure on $\mathbb{S}^{n-1}$ is the reciprocal of the Gauss curvature of $\partial K$.

For any $x\in\mathbb{R}^{n}$ we let $x^{\bot}$ be the $(n-1)$-subspace orthogonal to $x$, and let $P_{x^{\bot}}C$ be the orthogonal projection of $C$ onto $x^{\bot}$. Then the projection body $\Pi K$ of $K\in\mathcal{K}^{n}$ is the centrally symmetric convex body given by its support function

[TABLE]

for every $u\in\mathbb{S}^{n-1}$. Using standard properties of the mixed volume $V_{1}(\cdot,\cdot)$ (cf. [14]) it is easy to see that

[TABLE]

In fact,

[TABLE]

where $L_{u}=[-u,u]=\{tu:t\in[-1,1]\}$. Using Fubini’s formula,

[TABLE]

Then

[TABLE]

Finally, the polar projection body $\Pi^{\circ}K$ is the polar body of $\Pi K$.

A function $f:\mathbb{R}^{n}\rightarrow[0,\infty)$ is log-concave if $\log f$ is concave, i.e., if

[TABLE]

for every $x,y\in\mathbb{R}^{n}$, $\lambda\in(0,1)$. Then $f=e^{-\varphi}$ for a convex function $\varphi:\mathbb{R}^{n}\rightarrow[-\infty,\infty)$. Moreover, let $\mathcal{F}(\mathbb{R}^{n})=\{f\text{ log-concave with }f\in L^{1}(\mathbb{R}^{n})\}$.

Two typical embeddings of all convex bodies onto the set $\mathcal{F}(\mathbb{R}^{n})$ are given by the mappings that identify $K$ either with the characteristic function $\chi_{K}(x)=e^{-I^{\infty}_{K}(x)}$ or the exponential gauge $e^{-\|x\|_{K}}$ of $K$, where

[TABLE]

Considering the definition of $h_{K}$ given by (1), we can write

[TABLE]

where, for a convex function $\varphi$,

[TABLE]

is the so called Legendre transform of $\varphi$ (cf. [10]). As

[TABLE]

it is natural to define the support function of a log-concave function $f=e^{-\varphi}$ as

[TABLE]

(cf. [13]). Note that $h_{K}^{*}=I^{\infty}_{K}$ for every $K\in\mathcal{K}^{n}$ (cf. [10]).

In order to define the polar function of a function $f=e^{-\varphi}$ as a log-concave function, it is natural to search for a transformation $T$ between convex functions so that $f^{\circ}=e^{-T\varphi}$. Since $(K^{\circ})^{\circ}=K$ and if $K_{1}\subset K_{2}$ then $K_{1}^{\circ}\supset K_{2}^{\circ}$, we have to ask for $T$ to verify $T^{2}$ to be the identity, and if $\varphi_{1}\leq\varphi_{2}$, then $T\varphi_{1}\geq T\varphi_{2}$. From [5], these properties characterize the Legendre transform, so $T\varphi=\varphi^{*}$.

As a consequence, for any log-concave $f:\mathbb{R}^{n}\rightarrow[0,+\infty)$ with $f=e^{-\varphi}$, its polar function $f^{\circ}$ is defined by $f^{\circ}=e^{-\varphi^{*}}$ (cf. [5]). With this definition, if $f\in\mathcal{(}\mathbb{R}^{n})$ with $0\in\mathrm{int}(\mathrm{supp}f)$, and $f^{\circ}(x_{M})=\|f\|_{\infty}$ for some $x_{M}\in\mathbb{R}^{n}$, then $f^{\circ}\in\mathcal{F}(\mathbb{R}^{n})$ too (see Theorem 4.3 below). Note that $f^{\circ}=e^{-h_{f}}$.

To define the analogue definition of $\Pi f$ for a log-concave $f$, firstly defined in [9], we take into account the equality for a convex body $K$

[TABLE]

for $u\in\mathbb{S}^{n-1}$ (see [16], or Proposition 2.2 iii) ). We may now generalize (2) to define the Petty projection function $\Pi f$ of $f$ given its support function

[TABLE]

(see [9]). Note that, by the chain rule, if $f=e^{-\varphi}$, then $\nabla f=-f\nabla\varphi$, and the previous definition admits the form

[TABLE]

In particular, for any $f\in\mathcal{F}(\mathbb{R}^{n})$, the polar projection function is given by $\Pi^{\circ}f=(\Pi f)^{\circ}$.

2 Properties and main result

The main result here serves as a correction to [9, Thm. 5.2] and introduces a lower bound for the integral of $\Pi^{\circ}f$. Let us denote by $B^{n}_{2}$ the n-dimensional Euclidean unit ball, $\mathbb{S}^{n-1}$ its boundary, and let $\omega_{n}=\mathrm{vol}(B^{n}_{2})$ be its volume. Moreover, let $|x|=\sqrt{x_{1}^{2}+\cdots+x_{n}^{n}}$ be the Euclidean norm for every $x=(x_{1},\dots,x_{n})\in\mathbb{R}^{n}$.

Theorem 2.1**.**

Let $f\in\mathcal{F}(\mathbb{R}^{n})$. Then

[TABLE]

Moreover, equality holds if there exists $g:[0,\infty)\rightarrow[0,\infty)$, $g\in\mathcal{F}(\mathbb{R}^{1})$, such that $f(x)=g(|x|)$ for every $x\in\mathbb{R}^{n}$.

In the next proposition we collect some useful computations needed in this paper, refereed to characteristic functions and exponential gauges of convex bodies.

Proposition 2.2**.**

Let $K\in\mathcal{K}^{n}$. Then we have that

1. i)

$h_{\chi_{K}}=h_{K}$. 2. ii)

$h_{e^{-\|\cdot\|_{K}}}=I^{\infty}_{K^{\circ}}$. 3. iii)

$h_{\Pi\chi_{K}}=h_{\Pi K}$. 4. iv)

$\Pi\chi_{K}=e^{-I^{\infty}_{\Pi K}}=\chi_{\Pi K}$. 5. v)

$\Pi^{\circ}\chi_{K}=e^{-h_{\Pi K}}$. 6. vi)

$h_{\Pi e^{-\|\cdot\|_{K}}}=(n-1)!h_{\Pi K}$. 7. vii)

$\int_{\mathbb{R}^{n}}e^{-\|x\|_{K}}dx=n!\mathrm{vol}(K)=n!\int_{\mathbb{R}^{n}}\chi_{K}(x)dx$. 8. viii)

$\int_{\mathbb{R}^{n}}\Pi^{\circ}(e^{-\|\cdot\|_{K}})(x)dx=n!\Gamma(n)^{-n}\mathrm{vol}(\Pi^{\circ}K)=\Gamma(n)^{-n}\int_{\mathbb{R}^{n}}\Pi^{\circ}(\chi_{K})(x)dx$.

Let us observe that Theorem 5.2 in [9] is not correct. Indeed, using Proposition 2.2, one can verify that if $f=\chi_{tB^{n}_{2}}$, for some $t>0$, then the Theorem 5.2 in [9] becomes $(1-\log t)^{n-1}\leq C(n)$, for some constant $C(n)>0$ only depending on the dimension $n$. Later on, we will discuss how to correct those bounds for the integral of the Petty projection function.

One can also bound from above the term $\int|\nabla f|$ by means of an entropic function under some extra assumptions; indeed, if $a\chi_{B^{n}_{2}}\leq f$ for some $f\in\mathcal{F}(\mathbb{R}^{n})$ and $a>0$, then

[TABLE]

with equality if $f=a\chi_{B^{n}_{2}}$ (cf. [1], see also [6]).

The quantity $\mathrm{vol}(K)^{n-1}\mathrm{vol}(\Pi^{\circ}K)$ is an affine invariant, its maximum value is provided by Petty’s projection inequality [12], with equality if and only if $K$ is an ellipsoid, and its minimum value is given by Zhang’s inequality [15], with equality if and only if $K$ is a simplex:

[TABLE]

For any $f\in\mathcal{F}(\mathbb{R}^{n})$, $f=e^{-\varphi}$, let $\Pi_{b}f$ be the Petty projection body of $f$, which is the convex body whose support function is given by

[TABLE]

In order to avoid future confusion, here we have changed the original name also given by Fang and Zhou [9] (they used the name $\Pi f$, and we insert the subindex $b$ to stress that it is a body). Its polar $\Pi^{\circ}_{b}f$ was firstly introduced in [1], and here once more we change the old naming $\Pi^{\circ}f$ by $\Pi^{\circ}_{b}f$, and it is the unit ball of the norm given by

[TABLE]

Since $f=e^{-\varphi}$, due to $\nabla f(x)=-e^{-\varphi(x)}\nabla\varphi(x)$, we have that $\Pi^{\circ}_{b}f=(\Pi_{b}f)^{\circ}$, as one may expect. The right-hand side of (5) was extended to functional settings by Zhang [16] and it is also known as the affine Sobolev inequality, whereas the left-hand side of (5) was recently extended to log-concave functions in [2],

[TABLE]

Moreover, equality holds on the right-hand side if and only if $\frac{f}{\|f\|_{\infty}}=\chi_{AB^{n}_{2}}$, for any regular $A\in\mathbb{R}^{n\times n}$, and on the left hand side if and only if $\frac{f}{\|f\|_{\infty}}=e^{-\|\cdot\|_{S}}$, for any simplex $S\in\mathcal{K}^{n}$, with $0\in S$. One can immediately verify that

[TABLE]

and thus, (6) can be used to give optimal bounds of the integral of $\Pi^{\circ}f$ for any $f\in\mathcal{F}(\mathbb{R}^{n})$.

Proposition 2.3**.**

Let $f\in\mathcal{F}(\mathbb{R}^{n})$. Then

[TABLE]

After writing this note, Fang and Zhou have told us in personal communication that they also noticed their mistake; however, it seems that they have amended it replacing it by the right-hand side of (6) and using Proposition 2.3.

3 Proofs

We start this section by proving Proposition 2.2.

Proof of Proposition 2.2.

1. i)

See (3). 2. ii)

It is a direct consequence of $\|\cdot\|_{K}^{*}=I^{\infty}_{K^{\circ}}$. 3. iii)

Here we use a similar argument to the one exhibited in [16, §4]. Let us denote by $d(x,A)$ the Euclidean distance from a point $x\in\mathbb{R}^{n}$ to a set $A\subset\mathbb{R}^{n}$. Let $\varepsilon>0$ and define

[TABLE]

If $d(x,K)>0$ for some $x\in\mathbb{R}^{n}$, then there exists a unique $x^{\prime}\in\partial K$ such that $d(x,K)=|x-x^{\prime}|$. Let $\nu(x^{\prime})=\frac{x-x^{\prime}}{|x-x^{\prime}|}$ be the outer normal of $K$ at $x^{\prime}$ and let $D_{\varepsilon}=\{x\in\mathbb{R}^{n}:0<d(x,K)<\varepsilon\}$. Then

[TABLE]

from which

[TABLE]

When $\varepsilon\rightarrow 0$, we have that

[TABLE]

where $d\sigma(\partial K,\cdot)$ is the surface area element of $\partial K$. Since $\lim_{\varepsilon\rightarrow 0}f_{\varepsilon}=\chi_{K}$ and by (2) we can conclude that

[TABLE] 4. iv)

Since by definition $\Pi\chi_{K}(x)=e^{-h_{\Pi\chi_{K}}^{*}(x)}$, by iii) we can conclude that

[TABLE] 5. v)

Using iii) we have that $\Pi^{\circ}\chi_{K}(x)=e^{-h_{\Pi\chi_{K}}(x)}=e^{-h_{\Pi K}(x)}$, as desired. 6. vi)

See [9, Prop. 5.1]. 7. vii)

By definition, $\int_{\mathbb{R}^{n}}\chi_{K}(x)dx=\mathrm{vol}(K)$. Second,

[TABLE] 8. viii)

On the one hand, using v) and vii) we immediately get that

[TABLE]

On the other hand, using vi) and vii) and the 1-homogeneity of the support function we can conclude that

[TABLE]

∎

We now prove Theorem 2.1. The main ingredients of it are the integration by polar coordinates and the Jensen inequality [3], which states that if $(X,\Sigma,\mu)$ is a probability space, then for any convex function $\varphi:\mathbb{R}\rightarrow\mathbb{R}$ and any $\mu$-integrable function $f:X\rightarrow\mathbb{R}$, we have that

[TABLE]

and moreover, equality holds if and only if either $\varphi$ is affine or $f$ is independent of $x$. One can compare the proof below to the one in [9, Thm. 5.2], where we have detected mistakes in (5.17) at the change of variables and at the application of Jensen inequality.

Proof of Theorem 2.1.

Let $f=e^{-\varphi}$. Since $\nabla f(x)=-f(x)\nabla\varphi(x)$ and using polar coordinates, we can write

[TABLE]

where $\mu$ is the uniform probability measure in $\mathbb{S}^{n-1}$. Since $e^{x}$ is convex, Jensen inequality implies that

[TABLE]

Using Fubini and using the fact that

[TABLE]

for any $v\in\mathbb{R}^{n}$, then

[TABLE]

Letting $t=\frac{\omega_{n-1}}{n\omega_{n}}\|\nabla f\|_{1}r=ar$, then $dt=adr$ and

[TABLE]

We can thus conclude that

[TABLE]

In the equality case, there must be equality in the inequality above. Hence, by Jensen’s equality case, we must have that $\int_{\mathbb{R}^{n}}|\langle\nabla f(x),u\rangle|dx$ is independent of $u\in\mathbb{S}^{n-1}$. In particular, if $f(x)=g(|x|)$ for some $g:[0,\infty)\rightarrow[0,\infty)$ log-concave and every $x\in\mathbb{R}^{n}$, as desired. ∎

A geometrical consequence of Theorem 2.1 is the following result (cf. [16]), which relates the surface area measure $S(K)$ of a $K\in\mathcal{K}^{n}$ with the volume $\mathrm{vol}(\Pi^{\circ}K)$, and can be also obtained by Hölder inequality in (5).

Corollary 3.1**.**

Let $K\in\mathcal{K}^{n}$. Then

[TABLE]

Moreover, equality holds if $K=B^{n}_{2}$.

Proof.

Let us particularize Theorem 2.1 taking $f(x)=e^{-\|x\|_{K}}$. If we denote by $d\sigma(\partial K,\cdot)$ the surface area element of $K$, then

[TABLE]

This, together with viii) in Proposition 2.2, imply that

[TABLE]

as desired.

Equality holds if $e^{-\|x\|_{K}}$ is independent of $x\in\mathbb{S}^{n-1}$, for instance, if $K=B^{n}_{2}$. ∎

Now we show Proposition 2.3.

Proof of Proposition 2.3.

As a consequence of (7) and vii) in Proposition 2.2, we obtain that

[TABLE]

∎

4 Integrability of log-concave functions

In this section we characterize the integrability of log-concave functions in terms of the value of $f$ over all possible rays. Other characterizations of the integrability of log-concave functions were given in [8]. Before stating the next result, we would like to remember that for any log-concave function $f:\mathbb{R}^{n}\rightarrow[0,\infty)$, the function

[TABLE]

is log-concave, continuous on its support $\mathop{\mathrm{supp}}g=\mathop{\mathrm{supp}}f=\overline{\{x\in\mathbb{R}^{n}:f(x)>0\}}$, and has $\int g=\int f$ (see [7, Lem. 2.1]). Thus, we can always replace $f$ by $g$ and hence we can always extend continuously $f$ to its support.

Lemma 4.1**.**

Let $f:\mathbb{R}^{n}\rightarrow[0,\infty)$ be log-concave with $f(x_{M})=\|f\|_{\infty}$ for some $x_{M}\in\mathbb{R}^{n}$. Then the following are equivalent:

1. (1)

$f$* is integrable.* 2. (2)

Either $\int f=0$ or there exists no ray $R=x_{0}+\mathbb{R}_{+}u$, $x_{0},u\in\mathbb{R}^{n}$, $u\neq 0$, for which $f|_{R}=c>0$.

Proof.

We first prove (1) implies (2). Let us suppose that $\int f>0$. Hence, the function $f$ is non-zero in an open ball $B(x_{0},r)$, for some $x_{0}\in\mathbb{R}^{n}$ and $r>0$. By continuity of $f$ in $B(x_{0},r)$, let us suppose that $f(x)\geq\alpha$, for every $x\in B(x_{0},r)$ and for some $\alpha>0$. Moreover, for the sake of contradiction, let us suppose that there exists a ray $R=y_{0}+\mathbb{R}_{+}u$, $y_{0},u\in\mathbb{R}^{n}$, $u\neq 0$, such that $f|_{R}=c>0$. Let $z_{0}\in U=B(x_{0},r)\cap(x_{0}+u^{\bot})$ and $t>0$. Let us observe that

[TABLE]

Furthermore, we have that

[TABLE]

and thus that

[TABLE]

Note that $z_{0}+tu\in\mathrm{int}(\mathrm{supp}f)$. Hence

[TABLE]

thus showing that $f$ is not integrable, a contradiction.

We now show (2) implies (1). If $\int f=0$, then $f$ is integrable. Let us suppose that $\int f>0$. After a suitable translation, let us assume that $f(0)=\|f\|_{\infty}$. For every $u\in\mathbb{S}^{n-1}$, since $f$ is not constant on $\mathbb{R}_{+}u$, then there exists $s_{u}>0$ such that $f(s_{u}u)<\|f\|_{\infty}$. Now, using that $f$ is continuous implies that $t_{u}=\inf\{s>0:f(su)<\|f\|_{\infty}\}$ fulfills $f(t_{u}u)=\|f\|_{\infty}$, for every $u\in\mathbb{S}^{n-1}$. We now show that if $\{t_{u}:u\in\mathbb{S}^{n-1}\}$ is unbounded, we arrive at a contradiction. Indeed, in that case let $\{u_{k}\}\subset\mathbb{S}^{n-1}$ be such that $t_{u_{k}}\rightarrow\infty$ as $k\rightarrow\infty$. Since $\mathbb{S}^{n-1}$ is compact, there exists a subsequence (which we can suppose w.l.o.g. to be the sequence itself) converging $u_{k}\rightarrow u_{0}\in\mathbb{S}^{n-1}$. For every $t>0$, then

[TABLE]

Note that if $t>0$ is fixed, since $t_{u_{k}}\rightarrow\infty$, there exists $k_{t}\in\mathbb{N}$ such that if $k\geq k_{t}$ then $t_{u_{k}}\geq t$. Hence

[TABLE]

Therefore $f|_{\mathbb{R}_{+}u_{0}}=\|f\|_{\infty}$, contradicting the hypothesis. Thus $\{t_{u}:u\in\mathbb{S}^{n-1}\}$ is bounded. If $t>t_{u}$, then $f(tu)<\|f\|_{\infty}$. Hence let $t_{*}>t_{u}$ be such that $f(t_{*}u)\leq c<\|f\|_{\infty}$ for every $u\in\mathbb{S}^{n-1}$ and some $c>0$. Observe that for every $t\geq t_{*}$ and every $u\in\mathbb{S}^{n-1}$ we have that

[TABLE]

Thus, integrating in polar coordinates, we get that

[TABLE]

where $\mu$ is the uniform probability measure in $\mathbb{S}^{n-1}$. Since the first integral is finite as $0<c/f(0)<1$ and the second one is finite as $f$ is bounded and $B(0,t_{*})$ is bounded too, hence we obtain that $f$ is integrable. ∎

Remark 4.2**.**

The existence of $x_{M}\in\mathbb{R}^{n}$ in Lemma 4.1 is necessary. Indeed, the function $f:\mathbb{R}\rightarrow[0,\infty)$ where $f(x):=e^{-e^{x}}$, which is monotonically decreasing on $\mathbb{R}$, is log-concave, it fulfills (2) (since it is not constant over any ray) but does not fulfill (1) (simply noticing that $f(x)\geq e^{-1}$ for every $x\leq 0$).

Theorem 4.3**.**

Let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ be log-concave with $0\in\mathrm{int}(\mathrm{supp}(f))$ and $f^{\circ}(x_{M})=\|f^{\circ}\|_{\infty}$ for some $x_{M}\in\mathbb{R}^{n}$. Then $f^{\circ}$ is integrable.

Proof.

Let us suppose that $f=e^{-\varphi}$. We show now that $\varphi^{*}$ (and thus $f^{\circ}$) is not constant over any ray $y_{0}+\mathbb{R}_{+}u$, for any $y_{0},u\in\mathbb{R}^{n}$, $u\neq 0$, thus concluding by Lemma 4.1 that $f^{\circ}$ is integrable. Since $\varphi$ is continuous and $0\in\mathrm{int}(\mathrm{supp}f)$, let us suppose that $\varphi(x)\leq C$ whenever $|x|\leq\delta$ for some $C,\delta>0$. Let us consider $y_{0}+tu$, $t>0$. Then

[TABLE]

thus showing that $\varphi^{*}$ is not constant over any ray $y_{0}+\mathbb{R}_{+}u$, as desired. ∎

Remark 4.4**.**

The integrability of $f^{\circ}$ can be easily deduced by using some Blaschke-Santaló functional inequality

[TABLE]

but only for certain particular translations of $f$ (for instance, when the Santaló point of $f$ is the origin, see [4]). However, the comment in [10, Rmk. 2] is not correct (where the authors said that ”All of our results hold, with the same proofs, for log-concave functions that reach their maximum at the origin”), since $f^{\circ}$ is not necessarily integrable if $f(0)=\|f\|_{\infty}$. For instance, letting

[TABLE]

if $f=e^{-\varphi}$, then

[TABLE]

thus having that $f^{\circ}(x)=e^{0}=1$ if $x<0$, and hence $f^{\circ}$ would not be integrable.

Acknowledgements. We would like to thank Julián Haddad for his valuable comments and pointers, and to the University of Sevilla for hosting Leticia Alves da Silva during two months, in which this work has been done. She also thanks the support of the IFMG - Campus Bambuí while conducting this work.