Minkowski Symmetrizations of Star Shaped Sets

TL;DR

This paper establishes precise upper bounds on the number of Minkowski symmetrizations needed to approximate any star-shaped set in n-dimensional space by a Euclidean ball, measured via the Hausdorff metric.

Contribution

It provides the first sharp bounds for symmetrizations transforming star-shaped sets into near-spherical shapes in high-dimensional spaces.

Findings

Derived explicit bounds for symmetrizations in high dimensions

Showed convergence of star-shaped sets to Euclidean balls

Enhanced understanding of geometric symmetrization processes

Abstract

We provide sharp upper bounds for the number of symmetrizations required to transform a star shaped set in ${\mathbb R}^n$ arbitrarily close (in the Hausdorff metric) to the Euclidean ball.