The Sine Transform of Isotropic Measures

TL;DR

This paper establishes sharp isoperimetric inequalities for the sine transform of even isotropic measures, providing reverse inequalities and applications to volume estimates of convex bodies based on sections or projections.

Contribution

It introduces new sharp inequalities for the sine transform of isotropic measures and their reverse forms, with applications to convex geometry.

Findings

Sharp isoperimetric inequalities for the sine transform are proven.

Reverse inequalities are obtained in an asymptotically optimal form.

Applications include strong volume estimates for convex bodies from sections or projections.

Abstract

Sharp isoperimetric inequalities for the sine transform of even isotropic measures are established. The corresponding reverse inequalities are obtained in an asymptotically optimal form. These new inequalities have direct applications to strong volume estimates for convex bodies from data about their sections or projections.