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Journal of Convex Analysis 31 (2024), No. 2, 563--602
Copyright Heldermann Verlag 2024

About the Blaschke-Santaló Diagram of Area, Perimeter and Moment of Inertia

Raphael Gastaldello
Mines Nancy, Université de Lorraine CNRS, IECL, Nancy, France

Antoine Henrot
Mines Nancy, Université de Lorraine CNRS, IECL, Nancy, France

Ilaria Lucardesi
(1) Dip. di Matematica e Informatica "Ulisse Dini", University of Florence, Italy
(2) Dip. di Matematica, University of Pisa, Pisa, Italy

We study the Blaschke-Santaló 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 axes of symmetry. We discuss topological and geometrical properties of the diagram. As a by-product we address a conjecture by Pólya, in the simplified setting of double symmetry.

Keywords: Shape optimization, area, perimeter, moment of inertia, convex geometry, Blaschke-Santaló diagram.

MSC: 49Q10, 52A40, 28A75.

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