Related papers: Concavity principles for weighted marginals
Let $C$ be a pointed closed convex cone in $\mathbb{R}^n$ with vertex at the origin $o$ and having nonempty interior. The set $A\subset C$ is $C$-coconvex if the volume of $A$ is finite and $A^{\bullet}=C\setminus A$ is a closed convex set.…
By using Minkowski addition of convex functions, we prove convexity and rearrangement properties of solutions to some Hessian equations in $\R^3$ and Brunn-Minkowski and isoperimetric inequalities for related functionals.
We consider a general way to obtain Pr\'ekopa-Leindler and Borell-Brascamp-Lieb type inequalities from Brunn-Minkowski type inequalities and provide numerous examples. We use the same heuristic to prove a discrete version of the…
B\"or\"oczky, Lutwak, Yang and Zhang recently proved the log-Brunn-Minkowski inequality which is stronger than the classical Brunn-Minkowski inequality for two origin-symmetric convex bodies in the plane. This paper establishes the…
We characterize the restrictions of B\'ekoll\'e--Bonami weights of bounded hyperbolic oscillation, to subsets of the unit disc, thus proving an analogue of Wolff's restriction theorem for Muckenhoupt weights. Sundberg proved a discrete…
In this paper, we develop a basic theory of Orlicz affine and geominimal surface areas for convex and $s$-concave functions. We prove some basic properties for these newly introduced functional affine invariants and establish related…
The entropy power inequality, which plays a fundamental role in information theory and probability, may be seen as an analogue of the Brunn-Minkowski inequality. Motivated by this connection to Convex Geometry, we survey various recent…
In this paper, we establish some new Ostrowski's type inequalities for m- and (alpha,m)- logarithmically convex functions by using the Riemann-Liouville fractional integrals.
In this paper, we are interested in investigating a weighted variant of Hermite-Hadamard type inequalities involving convex functionals. The approach undertaken makes it possible to refine and reverse certain inequalities already known in…
A local classification of spacelike surfaces in Minkowski 4-space, which are invariant under spacelike rotations, and with mean curvature vector either vanishing or lightlike, is obtained. Furthermore, the existence of such surfaces with…
The aim of this paper is to study properties of sections of convex bodies with respect to different types of measures. We present a formula connecting the Minkowski functional of a convex symmetric body K with the measure of its sections.…
In this note we offer some log-concavity properties of certain functions related to Bessel functions of the first kind and modified Bessel functions of the first and second kind, by solving partially a recent conjecture on the…
Let $Z$ be a Boolean model based on a stationary Poisson process $\eta$ of compact, convex particles in Euclidean space ${\mathbb{R}}^d$. Let $W$ denote a compact, convex observation window. For a large class of functionals $\psi$, formulas…
We report recent advances on noncommutative martingale inequalities associated with convex functions. These include noncommutative Burkholder-Gundy inequalities associated with convex functions due to the present authors and Dirksen and…
In this paper we further develop the theory of f-divergences for log-concave functions and their related inequalities. We establish Pinsker inequalities and new affine invariant entropy inequalities. We obtain new inequalities on functional…
The aim of this paper is to develop the $L_p$ John ellipsoid for the geometry of log-concave functions. Using the results of the $L_p$ Minkowski theory for log-concave function established in \cite{fan-xin-ye-geo2020}, we characterize the…
We use Brunn-Minkowski inequalities for quermassintegrals to deduce a family of inequalities of Poincar\'e type on the unit sphere and on the boundary of smooth convex bodies in the $n$-dimensional Euclidean space.
A generalization of Young's inequality for convolution with sharp constant is conjectured for scenarios where more than two functions are being convolved, and it is proven for certain parameter ranges. The conjecture would provide a unified…
In the first part of the paper, we define an approximated Brunn-Minkowski inequality which generalizes the classical one for length spaces. Our new definition based only on distance properties allows us also to deal with discrete spaces.…
A recent study by Bojowald and Paily provided a path toward the identification of an effective quantum-spacetime picture of Loop Quantum Gravity, applicable in the "Minkowski regime", the regime where the large-scale (coarse-grained)…