Characterizations of the Euclidean Ball by Intersections with Planes and Slabs

Abstract

In this article we prove two characterizations of the Euclidean ball: (i) the only convex body in ${\mathbb{R}}^3$ such that every normal plane bisects the volume (or surface area) is the Euclidean ball, (ii) the only convex body K in ${\mathbb{R}}^3$ such that for two suitable values $h_1,$ $h_2,$ the surface area of the intersection between any slab of width h_i, and $\partial K$ is a constant $c_i,$ $i=1,2,$ is the Euclidean ball.

MSC:

• Primary 52A15, Secondary 52A38

Acknowledgment

The authors thank Iván González and Víctor Aguilar for their interesting comments on the topic of this article. We also thank the unknown referees for the suggestions which improved the exposition. This work was partially supported by UABC Movilidad Académica 2017.

Additional information

Notes on contributors

J. Jerónimo-Castro

JESÚS JERÓNIMO CASTRO received his Ph.D. in mathematics from the Centro de Investigación en Matemáticas (CIMAT). He has held visiting positions at University College London, ETH-Zentrum, and the Alfréd Rényi Institute. He has been a students’ trainer for the Mathematical Olympiads for more than 20 years. He likes to listen to music and enjoys reading elegant mathematical demonstrations very much.

C. Yee-Romero

CARLOS YEE-ROMERO received his Ph.D. in mathematics from CIMAT in 2004. Since then he has been working at the Universidad Autónoma de Baja California. His interests include geometry, topology, dynamical systems, and cosmology. When he is not doing math, he likes to ride his bicycle, walk in the hills, and try out new restaurants.