Steiner symmetrization with respect to the Kakutani-Fibonacci sequence of directions

TL;DR

This paper proves that successive Steiner symmetrizations of any planar measurable set with respect to the Kakutani-Fibonacci sequence of directions converge to a ball of the same measure, demonstrating a specific symmetrization process.

Contribution

It introduces and analyzes a new sequence of directions for Steiner symmetrization, showing convergence to a symmetric shape in the plane.

Findings

Successive symmetrizations converge to a ball of equal measure.

The Kakutani-Fibonacci sequence of directions ensures convergence.

The process applies to any planar measurable set of finite measure.

Abstract

In this paper we will prove that for any planar measurable set of finite measure $M$, its successive Steiner symmetrizations with respect to the Kakutani-Fibonacci sequence of directions converge to the ball $M^*$ centered at the origin and having the same measure.