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
raphael.gastaldello7@etu.univ-lorraine.fr
Antoine Henrot
Mines Nancy, Université de Lorraine CNRS, IECL, Nancy, France
antoine.henrot@univ-lorraine.fr
Ilaria Lucardesi
(1) Dip. di Matematica e Informatica "Ulisse Dini", University of Florence, Italy
(2) Dip. di Matematica, University of Pisa, Pisa, Italy
ilaria.lucardesi@unipi.it
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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