Gaussian Measures of Dilations of Convex Rotation- Ally Symmetric Sets in Cn

Gaussian Correlation Conjecture for Symmetric Convex Sets

Gaussian correlation conjecture states that the Gaussian measure of the intersection of two symmetric convex sets is greater or equal to the product of the measures. In this paper, firstly we prove that the inequality holds when one of the two convex sets is the intersection of finite centered ellipsoids and the other one is simply symmetric. Then we prove that any symmetric convex set can be a...

The Infinitesimal Form of Brunn-minkowski Type Inequalities

Log-Brunn-Minkowski inequality was conjectured by Boröczky, Lutwak, Yang and Zhang [7], and it states that a certain strengthening of the classical Brunn-Minkowski inequality is admissible in the case of symmetric convex sets. It was recently shown by Nayar, Zvavitch, the second and the third authors [27], that Log-Brunn-Minkowski inequality implies a certain dimensional Brunn-Minkowski inequal...

Gaussian measures of dilatations of convex symmetric sets

Fixed points for total asymptotically nonexpansive mappings in a new version of bead ‎space‎

The notion of a bead metric space is defined as a nice generalization of the uniformly convex normed space such as $CAT(0)$ space, where the curvature is bounded from above by zero. In fact, the bead spaces themselves can be considered in particular as natural extensions of convex sets in uniformly convex spaces and normed bead spaces are identical with uniformly convex spaces. In this paper, w...

A Geometric Approach to Correlation Inequalities in the Plane

By elementary geometric arguments, correlation inequalities for radially symmetric probability measures are proved in the plane. Precisely, it is shown that the correlation ratio for pairs of width-decreasing sets is minimized within the class of infinite strips. Since open convex sets which are symmetric with respect to the origin turn out to be width-decreasing sets, Pitt’s Gaussian correlati...