Related papers: Higher-Order Reverse Isoperimetric Inequalities fo…
We give a functional version of the affine isoperimetric inequality for log-concave functions which may be interpreted as an inverse form of a logarithmic Sobolev inequality inequality for entropy. A linearization of this inequality gives…
In this work, we extend a classical theorem of Keith Ball on integrals of log-concave functions along rays against the weight $r^{p-1}$ to the previously inaccessible regime $p\in (-1,0)$: if $g:\mathbb R^n\to\mathbb R_+$ is an integrable…
The inequality of Berwald is a reverse-H\"older like inequality for the $p$th average, $p\in (-1,\infty),$ of a non-negative, concave function over a convex body in $\mathbb{R}^n.$ We prove Berwald's inequality for averages of functions…
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…
In this paper, we establish mean width inequalities of sections and projections of convex bodies for isotropic measures with complete equality conditions, which extends the recent work of Alonso-Guti\'{e}rrez and Brazitikos. Different from…
We establishe an affine Hardy-Littlewood-Sobolev inequality concerning two different functions which is stronger than the classical Hardy-Littlewood-Sobolev inequality. Furthermore, we also prove reverse inequalities for the new…
We investigate geometric and functional inequalities for the class of log-concave probability sequences. We prove dilation inequalities for log-concave probability measures on the integers. A functional analogue of this geometric inequality…
Sharp affine fractional Sobolev inequalities for functions on $\mathbb R^n$ are established. For each $0<s<1$, the new inequalities are significantly stronger than (and directly imply) the sharp fractional Sobolev inequalities of Almgren…
The celebrated Petty's projection inequality is a sharp upper bound for the volume of the polar projection body of a convex body. Lutwak introduced the concept of mixed projection bodies and extended Petty's projection inequality.…
In this manuscript, we study the inequalities between measures of convex bodies implied by comparison of their projections and sections. Recently, Giannopoulos and Koldobsky proved that if convex bodies $K, L$ satisfy $|K|\theta^{\perp}|…
Sharp reverse affine isoperimetric inequalities for asymmetric Wulff shapes and their polars are established, along with the characterization of all extremals. These new inequalities have as special cases previously obtained simplex…
Mixed $f$-divergences, a concept from information theory and statistics, measure the difference between multiple pairs of distributions. We introduce them for log concave functions and establish some of their properties. Among them are…
There have been many parallel streams of research studying order isomorphisms of some specific sets $G$ of functions from a set $X$ to $\mathbb{R}\cup\{\pm\infty\}$, such as the sets of convex or Lipschitz functions. We develop in this…
The classical Petty projection inequality is an affine isoperimetric inequality which constitutes a cornerstone in the affine geometry of convex bodies. By extending the polar projection body to an inter-dimensional operator, Petty's…
The difference body operator enjoys different characterization results relying on its basic properties such as continuity, SL(n)-covariance, Minkowski valuation or symmetric image. The Rogers-Shephard and the Brunn-Minkowski inequalities…
We prove an isoperimetric inequality for the uniform measure on a uniformly convex body and for a class of uniformly log-concave measures (that we introduce). These inequalities imply (up to universal constants) the log-Sobolev inequalities…
In this paper we will provide a new proof of the fact that for any convex body $K\subseteq\R^n$ $$ \frac{{{2n}\choose{n}}}{n^n}n\int_0^\infty r^{n-1}\vol_n(K\cap(re_n+K))dr\leq\frac{(\vol_n(K))^{n+1}}{(\vol_{n-1}(P_{e_n^\perp}(K)))^n}, $$…
We extend the notion of John's ellipsoid to the setting of integrable log-concave functions. This will allow us to define the integral ratio of a log-concave function, which will extend the notion of volume ratio, and we will find the…
In this paper, we not only give the extensions of the results given in [7] by Gill et al. for log-convex functions, but also obtain some new Hadamard type inequalities for log-convex, m-convex and (alpha,m)-convex functions.
In this article, we prove that convex functions and log-convex functions obey certain general refinements that lead to several refinements and reverses of well known inequalities for matrices, including Young's inequality, Heinz inequality,…