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

In this paper we consider the so-called procedure of it Continuous Steiner Symmetrization, introduced by Brock in citebro95,bro00. It transforms every domain Omegasubsetsubsetℝd 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 γ-continuous map tmapstoOmegat, it can be slightly modified so to obtain the γ-continuity for a γ-dense class of domains Omega, namely, the class of polyedral sets in ℝd. This allows to obtain a sharp characterization of the Blaschke-Santal'o diagram of torsion and eigenvalue.