Related papers: Surface area measures of log-concave functions

In this paper, we extend a spherical variant of the Kowalski-S\{l}odkowski theorem due to Li, Peralta, Wang and Wang. As a corollary, we prove that every 2-local map in the set of all surjective isometries (without assuming linearity) on a…

Let $\{P_{\theta}:\theta \in {\mathbb R}^d\}$ be a log-concave location family with $P_{\theta}(dx)=e^{-V(x-\theta)}dx,$ where $V:{\mathbb R}^d\mapsto {\mathbb R}$ is a known convex function and let $X_1,\dots, X_n$ be i.i.d. r.v. sampled…

An inequality of Brascamp and Lieb provides a bound on the covariance of two functions with respect to log-concave measures. The bound estimates the covariance by the product of the $L^2$ norms of the gradients of the functions, where the…

For $0<p<\infty$ and $-2\le\alpha\le0$ we show that the $L^p$ integral mean on $rD$ of analytic function in the unit disk $D$ with respect to the weighted area measure $(1-|z|^2)^\alpha dA(z)$ is a logarithmically convex function of $r$ on…

We introduce canonical coordinates on minimal time-like surfaces in the n-dimensional Minkowski space and prove the existence and the uniqueness of these parameters. With respect to these coordinates the coefficients of the first…

We study coarea inequalities for metric surfaces -- metric spaces that are topological surfaces, without boundary, and which have locally finite Hausdorff 2-measure $\mathcal{H}^2$. For monotone Sobolev functions $u\colon X \to \mathbb{R}…

The purpose of this paper is to introduce the new concept of weighted floating functions associated with log concave or $s$-concave functions. This leads to new notions of weighted functional affine surface areas. Their relation to more…

We show that, for any prime power p^k and any convex body K (i.e., a compact convex set with interior) in Rd, there exists a partition of K into p^k convex sets with equal volume and equal surface area. We derive this result from a more…

In this paper, we explore the concept of $\sigma$-quasiconvexity for functions defined on normed vector spaces. This notion encompasses two important and well-established concepts: quasiconvexity and strong quasiconvexity. We start by…

In the first part Busemann concavity as non-negative curvature is introduced and a bi-Lipschitz splitting theorem is shown. Furthermore, if the Hausdorff measure of a Busemann concave space is non-trivial then the space is doubling and…

We consider a functional $\mathcal F$ on the space of convex bodies in $\R^n$ defined as follows: ${\mathcal F}(K)$ is the integral over the unit sphere of a fixed continuous functions $f$ with respect to the area measure of the convex body…

We introduce Blaschke addition and homothety operations on log-concave functions and study their affine-geometric consequences. Our starting point is the first variation formula of Falah and Rotem (Calc. Var. and PDE, 2026), which…

We prove new entropy inequalities for log concave and s-concave functions that strengthen and generalize recently established reverse log Sobolev and Poincare inequalities for such functions. This leads naturally to the concept of…

The support measures of a convex body are a common generalization of the curvature measures and the area measures. With respect to the Hausdorff metric on the space of convex bodies, they are weakly continuous. We provide a quantitative…

Motivated by a long-standing conjecture of Polya and Szeg\"o about the Newtonian capacity of convex bodies, we discuss the role of concavity inequalities in shape optimization, and we provide several counterexamples to the…

A complete classification of all zonal, continuous, and translation invariant valuations on convex bodies is established. The valuations obtained are expressed as principal value integrals with respect to the area measures. The convergence…

Functionals (i.e. functions of functions) are widely used in quantum field theory and solid-state physics. In this paper, functionals are given a rigorous mathematical framework and their main properties are described. The choice of the…

For $0<p<\infty$ and $\alpha\in (-\infty,\infty)$ we determine when the $L^p$ integral mean on $\{z\in\mathbb C: |z|\le r\}$ of an entire function with respect to the Gaussian area measure $e^{-\alpha|z|^2}\,dA(z)$ is logarithmic convex or…

Let $m(G)$ be the infimum of the volumes of all open subgroups of a unimodular locally compact group $G$. Suppose integrable functions $\phi_1 , \phi_2 \colon G \to [0,1]$ satisfy $\| \phi_1 \| \leq \| \phi_2 \|$ and $\| \phi_1 \| + \|…

We show that the fundamental objects of the $L_p$-Brunn-Minkowski theory, namely the $L_p$-affine surface areas for a convex body, are closely related to information theory: they are exponentials of R\'enyi divergences of the cone measures…