arXiv subjects

Umut Caglar

Publications and source records attributed to Umut Caglar.

5 recordsLinked to original sources

arXiv2020

Pinsker inequalities and related Monge-Ampère equations for log concave functions

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 affine surface area and lower and upper bounds for the Kullback-Leibler divergence in terms of functional affine surface area. The functional inequalities lead to new inequalities for L_p-affine surface areas for convex bodies.

math.DG↗

arXiv2016

Stability results for some geometric inequalities and their functional versions

The Blaschke Santaló inequality and the $L_p$ affine isoperimetric inequalities are major inequalities in convex geometry and they have a wide range of applications. Functional versions of the Blaschke Santaló inequality have been established over the years through many contributions. More recently and ongoing, such functional versions have been established for the $L_p$ affine isoperimetric inequalities as well. These functional versions involve notions from information theory, like entropy and divergence. We list stability versions for the geometric inequalities as well as for their functional counterparts. Both are known for the Blaschke Santaló inequality. Stability versions for the $L_p$ affine isoperimetric inequalities in the case of convex bodies have only been known in all dimensions for $p=1$ and for $p > 1$ only for convex bodies in the plane. Here, we prove almost optimal stability results for the $L_p$ affine isoperimetric inequalities, for all $p$, for all convex bodies, for all dimensions. Moreover, we give stability versions for the corresponding functional versions of the $L_p$ affine isoperimetric inequalities, namely the reverse log Sobolev inequality, the $L_p$ affine isoperimetric inequalities for log concave functions and certain divergence inequalities.

math.FA↗

arXiv2016

Mixed f-divergence and inequalities for log concave functions

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 affine invariant vector entropy inequalities, like new Alexandrov-Fenchel type inequalities and an affine isoperimetric inequality for the vector form of the Kullback Leibler divergence for log concave functions. Special cases of $f$-divergences are mixed $L_λ$-affine surface areas for log concave functions. For those, we establish various affine isoperimetric inequalities as well as a vector Blaschke Santaló type inequality.

math.FA↗

arXiv2016

Affine isoperimetric inequalities in the functional Orlicz-Brunn-Minkowski theory

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 functional affine isoperimetric inequalities as well as functional Santaló type inequalities.

math.MG↗

arXiv2013

Divergence for s-concave and log concave 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 f-divergence and, in particular, relative entropy for s-concave and log concave functions. We establish their basic properties, among them the affine invariant valuation property. Applications are given in the theory of convex bodies.

math.FA↗