Related papers: A reverse Rogers-Shephard inequality for log-conca…
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 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 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…
We fuse between the Rogers-Shephard inequality for the Lebesgue measure and Royen's Gaussian Correlation Inequality, simultaneously extending both into a single sharp inequality for the Gaussian measure $\gamma$ on $\mathbb{R}^n$, stating…
We present a short proof of the Alexandrov-Fenchel inequalities for mixed volumes of convex bodies.
We prove a Santal\'{o} and a reverse Santal\'{o} inequality for the polarity transform, which was recently re-discovered by Artstein-Avidan and Milman, in the class consisting of (even) log-concave functions attaining their maximal value 1…
Zhang's reverse affine isoperimetric inequality states that among all convex bodies $K\subseteq\mathbb{R}^n$, the affine invariant quantity $|K|^{n-1}|\Pi^*(K)|$ (where $\Pi^*(K)$ denotes the polar projection body of $K$) is minimized if…
We prove that the reciprocal of the volume of the polar bodies, about the Santal\'o point, of a {\em shadow system} of convex bodies $K_t$, is a convex function of $t$. Thus extending to the non-symmetric case a result of Campi and Gronchi.…
On the class of log-concave functions on $\R^n$, endowed with a suitable algebraic structure, we study the first variation of the total mass functional, which corresponds to the volume of convex bodies when restricted to the subclass of…
Euler's gamma function is logarithmically convex on positive semi-axis. Additivity of logarithmic convexity implies that the function sum of gammas with non-negative coefficients is also log-convex. In this paper we investigate the series…
The Wills functional $\mathcal{W}(K)$ of a convex body $K$, defined as the sum of its intrinsic volumes $\mathrm{V}_i(K)$, turns out to have many interesting applications and properties. In this paper we make profit of the fact that it can…
We establish the log-concavity of the volume of central sections of dilations of the cross-polytope (the strong B-inequality for the cross-polytope and Lebesgue measure restricted to an arbitrary subspace).
We establish new functional versions of the Blaschke-Santal\'o inequality on the volume product of a convex body which generalize to the non-symmetric setting an inequality of K. Ball and we give a simple proof of the case of equality. As a…
In this note prove the following Berwald-type inequality, showing that for any integrable log-concave function $f:\mathbb R^n\rightarrow[0,\infty)$ and any concave function $h:L\rightarrow\mathbb [0,\infty)$, where $L$ is the epigraph of…
We introduce the notion of Loewner (ellipsoid) function for a log concave function and show that it is an extension of the Loewner ellipsoid for convex bodies. We investigate its duality relation to the recently defined John (ellipsoid)…
We prove, using optimal transport tools, weighted Poincar'e inequalities for log-concave random vectors satisfying some centering conditions. We recover by this way similar results by Klartag and Barthe-Cordero-Erausquin for log-concave…
In this paper we extend some notions, previously defined for log-concave functions, to the larger domain of so-called {\alpha}-concave functions. We begin with a detailed discussion of support functions - first for log-concave functions,…
We study convexity or concavity of certain trace functions for the deformed logarithmic and exponential functions, and obtain in this way new trace inequalities for deformed exponentials that may be considered as generalizations of…
In this note we examine the volume of the convex hull of two congruent copies of a convex body in Euclidean $n$-space, under some subsets of the isometry group of the space. We prove inequalities for this volume if the two bodies are…
A sharp quantitative version of the $L_p-$mixed volume inequality is established. This is achieved by exploiting an improved Jensen inequality. This inequality is a generalization of Pinsker-Csisz\'ar-Kullback inequality for the Tsallis…