Blaschke's rolling ball theorem and the Trudinger-Wang monotone bending

TL;DR

This paper connects Blaschke's classical theorem on convex surfaces with positive curvature to the Trudinger-Wang optimal transport theory, and discusses an application in reflector antenna design.

Contribution

It reveals a new link between convex geometry and optimal transport theory, extending classical results to modern applications.

Findings

Established a connection between Blaschke's theorem and Trudinger-Wang's inclusion principle.

Applied the theoretical link to reflector antenna design.

Provided insights into convex surface properties and optimal transport applications.

Abstract

We revisit the classical rolling ball theorem of Blaschke for convex surfaces with positive curvature and show that it is linked to another inclusion principle in the optimal mass transportation theory due to Trudinger and Wang. We also discuss an application to reflector antennae design problem.