The Log-brunn-minkowski Inequality

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...

The Brunn-Minkowski-Type Inequality

Borell’s generalized Prékopa-Leindler inequality: A simple proof

We present a simple proof of Christer Borell’s general inequality in the Brunn-Minkowski theory. We then discuss applications of Borell’s inequality to the log-Brunn-Minkowski inequality of Böröczky, Lutwak, Yang and Zhang.

On the Orlicz-Brunn-Minkowski theory

Recently, Gardner, Hug and Weil developed an Orlicz-Brunn1 Minkowski theory. Following this, in the paper we further consider the 2 Orlicz-Brunn-Minkowski theory. The fundamental notions of mixed quer3 massintegrals, mixed p-quermassintegrals and inequalities are extended to 4 an Orlicz setting. Inequalities of Orlicz Minkowski and Brunn-Minkowski 5 type for Orlicz mixed quermassintegrals are o...

Volume difference inequalities for the projection and intersection bodies

In this paper, we introduce a new concept of volumes difference function of the projection and intersection bodies. Following this, we establish the Minkowski and Brunn-Minkowski inequalities for volumes difference function of the projection and intersection bodies.

Dimensional behaviour of entropy and information

We develop an information-theoretic perspective on some questions in convex geometry, providing for instance a new equipartition property for log-concave probability measures, some Gaussian comparison results for log-concave measures, an entropic formulation of the hyperplane conjecture, and a new reverse entropy power inequality for log-concave measures analogous to V. Milman’s reverse Brunn-M...