An application of the continuous Steiner symmetrization to Blaschke-Santaló diagrams

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-Santaló diagram of torsion and eigenvalue.

arXiv comment11 pages, 3 figures