An application of the continuous Steiner symmetrization to Blaschke-Santal\'o diagrams

TL;DR

This paper explores the continuous Steiner symmetrization process, modifying it for polyhedral domains to characterize the Blaschke-Santaló diagram relating torsion and eigenvalues.

Contribution

It introduces a modified symmetrization method that achieves gamma-continuity for polyhedral domains, enabling a sharp characterization of the Blaschke-Santaló diagram.

Findings

Modified symmetrization achieves gamma-continuity for polyhedral sets.

Provides a sharp characterization of the Blaschke-Santaló diagram.

Links symmetrization with spectral and torsional properties of domains.

Abstract

In this paper we consider the so-called procedure of {\it Continuous Steiner Symmetrization}, introduced by Brock in \cite{bro95,bro00}. It transforms every domain $\Omega\subset\subset\mathbb{R}^d$ into the ball keeping the volume fixed and letting the first eigenvalue and the torsion respectively decrease and increase. While this does not provide, in general, a $\gamma$-continuous map $t\mapsto\Omega_t$, it can be slightly modified so to obtain the $\gamma$-continuity for a $\gamma$-dense class of domains $\Omega$, namely, the class of polyedral sets in $\mathbb{R}^d$. This allows to obtain a sharp characterization of the Blaschke-Santal\'o diagram of torsion and eigenvalue.