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

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 $Omegasubsetsubsetmathbb{R}^d$ 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 $tmapstoOmega_t$, 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-Santalo diagram of torsion and eigenvalue.