Measures of Convex Bodies with Caffarelli Log-Concave Perturbation Theorem and Brunn–Minkowski Inequalities

Abstract

We study the sections , eestimates for the affine , dual affine quermassintegrals, slicing inequalities for measures and estimates for measures of lower dimensional sections of convex bodies in addition the boundary regularity of maps with convex potentials . The centroid bodies, logarithmic Laplace transform, monotonicity properties of optimal transportation ,rigidity , stability of caffarellis log-concave perturbation theorem and related inequalities examined and characterized . The behavior of the extensions of the Brunn-Minkowski and Prbkopa-Leindler theorems, including inequalities for log concave functions, and application to the diffusion equation are obtained . We give the relations form Brunn Minkowski to brascamp and to sharp and logarithmic sobolev inequalities. We conclude the study by the stability ,Gaussian and logarithmic Brunn–Minkowski type inequalities.