An affine isoperimetric inequality for log-concave functions^∗

The isoperimetric problem was known in Ancient Greece. However, the first mathematically rigorous proof was obtained only in the 19th century by Weierstrass based on works of Bernoulli, Euler, Lagrange and others. The isoperimetric problem is equivalent to the isoperimetric inequality that is bounding by the surface area and volume of the geometric domain $K$ in the Euclidian space $\mathbb{R}^{n}$ with equality if and only if $K$ is a standard ball. The natural generalizations of the isoperimetric inequality are Alexandrov-Fenchel inequalities that are bounding by mixed volumes of convex bodies in integral and convex geometry analysis. Blascheke-Santaló inequality, differential affine isoperimetric inequality, Busemann-Petty centroid inequality, Petty projection inequality and more affine isoperimetric inequalities are found with equalities for ellipsoids, polar bodies or simplices [8, 14, 15, 27, 34, 35, 36, 37, 39, 41, 46].

During past decades, the reverse affine isoperimetric inequalities, with simplices, cubes or polar bodies as extremal, received more attentions. Usually, reverse affine isoperimetric inequalities are much harder to establish. One of the known open problem about the reverse affine isoperimetric inequalities is the Mahler conjecture.

Let $\mathcal{K}_{o}^{n}$ denote the set of compact convex sets that contain the origin in their interiors in $\mathbb{R}^{n}$. Let $K\in\mathcal{K}_{o}^{n}$, the polar body $K^{\circ}$ of $K$, is defined by

and $x\cdot y$ denotes the inner product of $x$ and $y$.

If $K\in\mathcal{K}_{o}^{n}$ is symmetric, Mahler[40] conjectured that

where $|K|$ denotes the volume of $K$, with equality if and only if either $K$ or $K^{\circ}$ is a parallelepiped.

Inequality (1.2) was proved by Mahler [40] for $n=2$, the $3$-dimensional case was solved by Iriyeh and Shibata [28], and the case when $n\geq 4$ is still open. One can see [8, 9, 10, 14, 15, 25, 26, 27, 34, 35, 36, 37, 38, 39, 41, 46, 47, 48, 49] for more references on Mahler’s conjecture, affine isoperimetric inequalities and reverse affine isoperimetric inequalities.

Let $K\in\mathcal{K}_{o}^{n}$. The radial function of $\Gamma_{-2}K$, usually called the LYZ ellipsoid, is defined by

where $n_{K}(x)$ denotes the outer unit normal vector of at $x\in\partial K$, $h_{K}$ is the support function of $K$ and $\mathcal{H}^{n-1}$ is the $(n-1)$-dimensional Hausdorff measure. Another known reverse affine isoperimetric inequality is the following LYZ polar inequality [38].

LYZ polar inequality. Let $K\in\mathcal{K}_{o}^{n}$, then

with equality if and only if $K$ is a simplex whose centroid is at the origin.

Recently, Fang and Zhou [20] introduced the LYZ ellipsoid of log-concave functions by solving an extremum problem. A function $f:\mathbb{R}^{n}\to\mathbb{R}$ is called a log-concave function if $\log f$ is concave. Log-concave functions would be considered as extension of convex bodies. Usually, the results of convex bodies can be covered by taking log-concave functions, such as the Gaussian function or the characteristic function of convex bodies. The researches on log-concave functions originated from Ball’s Phd thesis [7], in which the Blaschke-Santaló inequality of even log-concave function was established. In recent years, some considerable progress has been made in establishing the geometric inequalities of log-concave function. Colesanti and Fragalà[17] established the Minkowski inequality of log-concave function via the Prékopa-Leindler inequality. Fang, Xing and Ye[19] further gave $L_{p}$ Minkowski inequalities and solved the existence of the even $L_{p}$ Minkowski problem of log-concave functions for $p>1$. John and Löwner ellipsoids for log-concave functions and their related inequalities were given in [3, 32]. Some affine isoperimetric inequalities of the affine surface areas of log-concave functions were obtained in [12, 13]. Other inequalities, such as Pólya-Szegö inequality, Rogers-Shephard inequality and its reverse and some stability results for log-concave functional inequalites can be found in [1, 2, 3, 4, 5, 6, 10, 11, 16, 22, 23, 29, 30, 31, 32, 33].

In this paper, for a convex function $\varphi:\mathbb{R}^{n}\to\mathbb{R}\cup\{\infty\}$, we use $\varphi^{\ast}$ to denote the Legendre transform of $\varphi$, and $f^{\circ}=e^{-\varphi^{\ast}}$ to denote the dual of a log-concave function $f$. The integral of $f$ on $\mathbb{R}^{n}$ is denoted by $J(f)$ and the domain of $\varphi$ is denoted by ${\rm dom}(\varphi)=\{x:\varphi(x)<\infty\}$. For any function $f:\mathbb{R}^{n}\to\mathbb{R}$ and $t\in\mathbb{R}$, the upper level set of $f$ is defined by

Let ${\rm LC}_{n}$ denote the class of all upper semi-continuous log-concave functions $f=e^{-\varphi}$ with ${\rm dom}(\varphi)=\mathbb{R}^{n}$ and $0<J(f)<\infty$.

Let $f\in\ {\rm LC}_{n}$, for the LYZ ellipsoid $\Gamma_{-2}f$ of the log-concave function $f$, defined in Definition 4.1, we have the following affine isoperimetric inequality, the functional LYZ polar inequality of (1.4).

Theorem. Let $f=e^{-\varphi}\in{\rm LC}_{n}$ and $\mu_{f}$ be the surface area measure of $f$. Suppose that $\varphi^{\ast}(x)>0$ on $\mathbb{R}^{n}\backslash\{o\}$ and $-\log t=\sup\{\varphi^{\ast}(x/\varphi^{\ast}(x)):x\in{\rm supp}(\mu_{f})\}$, then

Moreover, if $\varphi(x)>\varphi(o)$ for all $x\in\mathbb{R}^{n}\backslash\{o\}$, then equality holds if and only if $f(x)=e^{-{\|x\|_{K}}+1}$ and $K$ is a simplex whose centroid is at the origin.

Note that the inequality (1.5) is affine invariant, that is, the inequality (1.5) does not change under compositions of $f$ with affine transformations of $\mathbb{R}^{n}$. Inequality (1.5) includes the LYZ polar inequality as a special case. In particular, when $f(x)=e^{-{\|x\|_{K}}+1}$ and $K$ is a convex body contains origin in its interior, the inequality (1.5) becomes the LYZ polar inequality (1.4). A detailed derivation can be found in Section 5. The supremum $-\log t$ may achieve infinity and in this case the inequality (1.5) is strict. However, $-\log t$ is often bounded and some examples of the function $f$ will be given in Section 5.

This paper is organized as follows. In Section 2, we provide some basics of convex bodies and log-concave functions. The Ball-Barth inequality for isotropic embeddings, which is a key tool to establish our main theorem, is introduced in Section 3. In Section 4, the definition of the LYZ ellipsoid of the log-concave function $\Gamma_{-2}f$ is given, and its properties is listed. In Section 5, we give the proof of the main theorem and show that LYZ’s polar inequality is a direct result of inequality (1.5) ((Theorem 5.1).

In this section, we list some preliminaries and notations about convex bodies and log-concave functions. Good general references for the theory of convex bodies and log-concave functions are provided by the books of Gardner [24], Schneider[45] and Rockfellar [43].

Let $\mu$ be a positive Borel measure $\mu$ on $S^{n}$. A positive semi-definite $n\times n$ matrix $[\mu]$ can be generated by $\mu$, which is defined by

where $u\otimes u$ is the rank $1$ matrix generated by $u\in S^{n}$, or, equivalently by

for all $v\in S^{n}$. A positive Borel measure $\mu$ is said to be an isotropic measure if

where $I_{n}$ is the identity matrix. If $\mu$ is isotropic, summing the equation (3.2) with $v=e_{i},i=1,\cdots,n+1$, one has

The following Ball-Barth inequality was established in [36]. The Ball-Barth inequality. Let $\mu$ an isotropic measure on $S^{n}$ and $l:S^{n}\rightarrow(0,\infty)$ be a continuous function. Then

with equality if and only if $l(u_{1})\cdots l(u_{n+1})$ is a constant for linearly independent $u_{1},\cdots,u_{n+1}$ in ${\rm supp}(\nu)$.

The isotropic embedding from $\mathbb{R}^{n}$ to $S^{n}$ is given as follows.

Let $\mu$ and $h$ have been given in Definition 3.1. Summing the equation (3.6) with $u=e_{1},\cdots,e_{n},e_{n+1}$, one has

Define a new Borel measure $\nu$ on $S^{n}$, which is the push-forward of $\mu$ by $h$, by

for any Borel function $g:S^{n}\to\mathbb{R}$. Definition 3.1 shows that $\nu$ is an isotropic measure on $S^{n}$, by applying the Ball-Barthe inequality to $\nu$, we can obtain the following Ball-Barthe inequality for isotropic embeddings.

In this section, we will define the LYZ ellipsoid of log-concave functions, and introduce its properties.

Note that the function $\sqrt{-\log\Gamma_{-2}f(x)}$ is a Minkowski functional of an ellipsoid. Indeed, if $E$ is an ellipsoid generated by $n\times n$ real symmetric matrix $A$, that is

one can check that the Minkowski function of $E$ is given by $\|x\|_{E}^{2}=x\cdot Ax.$ From this, we see that the function $\sqrt{-\log\Gamma_{-2}f(x)}$ is a Minkowski functional of an ellipsoid, which is generated by $n\times n$ real symmetric matrix $M=[m_{ij}(f)]$, where

The following corollary shows that the quantity $|K_{t}(f^{\circ})|J(\Gamma_{-2}(f))$ is invariant under the operation of $GL(n)$.

When $f=e^{-\frac{\|x\|_{K}^{2}}{2}}$ with $K\in\mathcal{K}_{o}^{n}$, $\Gamma_{-2}f(x)$ can be calculated precisely.

When $f$ is the Gaussian function $e^{-\frac{|x|^{2}}{2}}$, the following result can be obtained directly from Property 4.2 and the fact that $\Gamma_{-2}B^{n}=B^{n}$.

In this section, we will establish the LYZ polar inequality for log-concave functions. The following lemmas are needed.

Note that $-\log t$ may reach infinity, but it is often bounded for many log-concave functions, such as $f=e^{-\varphi}\in{\rm LC}_{n}$ with $\varphi^{\ast}(x)>0$ for $x\in\mathbb{R}^{n}\backslash\{o\}$ and ${\rm dom}(\varphi^{\ast})=\mathbb{R}^{n}$. We first need to show $J(f^{\circ})<\infty$. For the Legendre transformation of $\varphi$,

we have $\varphi^{\ast}(y)\to\infty$ as $|y|\to\infty$ and hence $J(f^{\circ})<\infty$. Moreover, $f^{\circ}\in{\rm LC}_{n}$. [17, Lemma 2.5] shows that there exists two constants $a>0$ and $b$ such that

for any $x\in\mathbb{R}^{n}$. Hence

that is, for any $\varepsilon>0$, there exists a constant $M$ such that ${x}/{\varphi^{\ast}(x)}\in\left(\frac{1}{a}+\varepsilon\right)B^{n}$ for $x\in(MB^{n})^{c}$ and the unit ball $B^{n}$ in $\mathbb{R}^{n}$. Then $\varphi^{\ast}\left({x}/{\varphi^{\ast}(x)}\right)$ is bounded for any $x\in(M_{0}B^{n})^{c}$ as the continuity of $\varphi^{\ast}$. Set $m=\min\{\varphi^{\ast}(x):x\in MB^{n}\}$, then $x/\varphi^{\ast}(x)\in(M/m)B^{n}$ for any $x\in MB^{n}$. Hence $\varphi^{\ast}\left({x}/{\varphi^{\ast}(x)}\right)$ is bounded on $MB^{n}$ and $-\log t$ is a finite constant.

The following lemma is the key for the equality condition of our main inequality.