Log-Concave Functions

Abstract

We attempt to provide a description of the geometric theory of log-concave functions. We present the main aspects of this theory: operations between log-concave functions; duality; inequalities including the Prékopa-Leindler inequality and the functional form of Blaschke-Santaló inequality and its converse; functional versions of area measure and mixed volumes; valuations on log-concave functions.

Appendices

Appendix A. Basic Notions of Convex Geometry

This part of the paper contains some notions and constructions of convex geometry that are directly invoked throughout this paper. Our main reference text on the theory of convex bodies is the monograph [55].

A.1 Convex Bodies and Their Dimension

We denote by $\mathcal{K}^{n}$ the class of convex bodies, i.e. compact convex subsets of $\mathbb{R}^{n}$.

Given a convex body K its dimension is the largest integer k ∈ { 0, …, n} such that there exists a k-dimensional hyperplane of $\mathbb{R}^{n}$ containing K. In particular, if K has non-empty interior, then its dimension is n. The relative interior of K is the set of points x ∈ K such that there exists a k-dimensional ball centered at x included in K, where k is the dimension of K. If the dimension of K is n, then the relative interior coincides with usual interior.

A.2 Minkowski Addition

The Minkowski linear combination of $K,L \in \mathcal{K}^{n}$ with coefficients α, β ≥ 0 is

$$\displaystyle{\alpha K +\beta L =\{\alpha x +\beta y\,:\, x \in K,\,y \in L\}.}$$

It is easy to check that this is still a convex body.

A.3 Support Function

The support function of a convex body K is defined as:

$$\displaystyle{h_{K}\,:\, \mathbb{R}^{n} \rightarrow \mathbb{R},\quad h_{ K}(x) =\sup _{y\in K}(x,y).}$$

This is a 1-homogeneous convex function in $\mathbb{R}^{n}$. Vice versa, to each 1-homogeneous convex function h we may assign a unique convex body K such that h = h _K . Support functions and Minkowski additions interact in a very simple way; indeed, for every K and L in $\mathcal{K}^{n}$ and α, β ≥ 0 we have

$$\displaystyle{h_{\alpha K+\beta L} =\alpha h_{K} +\beta h_{L}.}$$

A.4 Hausdorff Metric

$\mathcal{K}^{n}$ can be naturally equipped with a metric: the Hausdorff metric d _H . One way to define d _H is as the $L^{\infty }(\mathbb{S}^{n-1})$ distance of support functions, restricted to the unit sphere:

$$\displaystyle{d_{H}(K,L) =\| h_{K} - h_{L}\|_{L^{\infty }(\mathbb{S}^{n-1})} =\max \{ \vert h_{K}(x) - h_{L}(x)\vert \,:\, x \in \mathbb{S}^{n-1}\}.}$$

Hausdorff metric has many useful properties; in particular, we note that $\mathcal{K}^{n}$ is a locally compact space with respect to d _H .

A.5 Intrinsic Volumes

An easy way to define intrinsic volumes of convex bodies is through the Steiner formula. Let K be a convex body and let B _n denote the closed unit ball of $\mathbb{R}^{n}$. For ε > 0 the set

$$\displaystyle{K +\epsilon B_{n} =\{ x +\epsilon y\,:\, x \in K,\,y \in B\} =\{ y \in \mathbb{R}^{n}\,:\,\mathrm{ dist}(x,K) \leq \epsilon \}}$$

is called the parallel set of K and denoted by K _ε . The Steiner formula asserts that the volume of K _ε is a polynomial in ε. The coefficients of this polynomial are, up to dimensional constants, the intrinsic volumes V ₀(K), …, V _n (K) of K:

$$\displaystyle{V _{n}(K_{\epsilon }) =\sum _{ i=0}^{n}V _{ i}(K)\epsilon ^{n-i}\kappa _{ n-i}.}$$

Here κ _j denotes the j-dimensional volume of the unit ball in $\mathbb{R}^{j}$, for every $j \in \mathbb{N}$. Among the very basic properties of intrinsic volumes, we mention that: V ₀ is constantly 1 for every K; V _n is the volume; V _n−1 is (n − 1)-dimensional Hausdorff measure of the boundary (only for those bodies with non-empty interior). Moreover, intrinsic volumes are continuous with respect to Hausdorff metric, rigid motion invariant, monotone, and homogeneous with respect to dilations (V _i is i-homogeneous). Finally, each intrinsic volume is a valuation

$$\displaystyle{ V _{i}(K \cup L) + V _{i}(K \cap L) = V _{i}(K) + V _{i}(L) }$$

(41)

for every K and L in $\mathcal{K}^{n}$, such that $K \cup L \in \mathcal{K}^{n}$. Hadwiger’s theorem claims that every rigid motion invariant and continuous valuation can be written as the linear combination of intrinsic volumes.

A.6 Mixed Volumes

The Steiner formula is just an example of the polynomiality of the volume of linear combinations of convex bodies. A more general version of it leads to the notions of mixed volumes. Let $m \in \mathbb{N}$ and K ₁, …, K _m be convex bodies; given λ ₁, …, λ _m  ≥ 0, the volume of the convex body λ ₁ K ₁ + … +λ _m K _m is a homogeneous polynomial of degree n in the variables λ _i ’s, and its coefficients are the mixed volumes of the involved bodies. The following more precise statement is a part of Theorem 5.16 in [55]. There exists a function $V \,:\, (\mathcal{K}^{n})^{n} \rightarrow \mathbb{R}_{+}$, the mixed volume, such that

$$\displaystyle{V _{n}(\lambda _{1}K_{1} +\ldots +\lambda _{m}K_{m}) =\sum _{ i_{1},\ldots,i_{n}=1}^{m}\lambda _{ i_{1}}\cdots \lambda _{i_{n}}V (K_{i_{1}},\ldots,K_{i_{n}})}$$

for every K ₁, …, K _m  ∈ K ⁿ and λ ₁, …, λ _m  ≥ 0. Hence a mixed volume is a function of n convex bodies. Mixed volumes have a number of interesting properties. In particular they are non-negative, symmetric, and continuous; moreover, they are linear and monotone with respect to each entry.

A.7 The Polar Body

The polar of a convex body K, having the origin as an interior point, is the set

$$\displaystyle{K^{\circ } =\{ y\,:\, (x,y) \leq 1\;\quad \forall x \in K\}.}$$

This is again a convex body, with the origin in its interior, and (K ^∘)^∘ = K.