Billiards, scattering by rough obstacles, and optimal mass transportation

Abstract

This article presents a brief exposition of recent results of the author on billiard scattering by rough obstacles. We define the notion of a rough body and give a characterization of scattering by rough bodies. Then we define the resistance of a rough body; it can be interpreted as the aerodynamic resistance of the somersaulting body moving through a rarefied medium. We solve the problems of maximum and minimum resistance for rough bodies (more precisely, for bodies obtained by roughening a prescribed convex set) in arbitrary dimension. Surprisingly, these problems are reduced to special problems of optimal mass transportation on the sphere.