About the Blaschke-Santalo diagram of area, perimeter and moment of inertia

TL;DR

This paper investigates the geometric relationships between area, perimeter, and moment of inertia in convex, doubly symmetric shapes in 2D, analyzing the Blaschke-Santaló diagram and addressing Pólya's conjecture.

Contribution

It characterizes the topological and geometrical properties of the Blaschke-Santaló diagram for convex, doubly symmetric shapes and provides insights into Pólya's conjecture within this setting.

Findings

Characterized the shape of the Blaschke-Santaló diagram under given assumptions.

Proved certain topological properties of the diagram.

Addressed Pólya's conjecture in the context of doubly symmetric convex shapes.

Abstract

We study the Blaschke-Santal\'o diagram associated to the area, the perimeter, and the moment of inertia. We work in dimension 2, under two assumptions on the shapes: convexity and the presence of two orthogonal axis of symmetry. We discuss topological and geometrical properties of the diagram. As a by-product we address a conjecture by P\'olya, in the simplified setting of double symmetry.