On Blaschke-Santal\'o diagrams for the torsional rigidity and the first Dirichlet eigenvalue

TL;DR

This paper investigates the geometric properties of Blaschke-Santaló diagrams relating torsional rigidity and the first Dirichlet eigenvalue under convexity and volume constraints, exploring their topology, shape, and conjectures.

Contribution

It provides a detailed analysis of the topological and geometric features of these diagrams, including conjectures about their boundary shapes and slopes.

Findings

Diagrams are studied under convexity and volume constraints.

Topological properties like closedness and simply connectedness are discussed.

Shape and slope properties near the ball are analyzed.

Abstract

We study Blaschke-Santal\'o diagrams associated to the torsional rigidity and the first eigenvalue of the Laplacian with Dirichlet boundary conditions. We work under convexity and volume constraints, in both strong (volume exactly one) and weak (volume at most one) form. We discuss some topological (closedness, simply connectedness) and geometric (shape of the boundaries, slopes near the point corresponding to the ball) properties of these diagrams, also providing a list of conjectures.