Convexity of the radial sum of a star body and a ball

Sakata, Shigehiro 

Canadian mathematical bulletin, Tome 67 (2024) no. 2, pp. 289-303

Résumé

We investigate the convexity of the radial sum of two convex bodies containing the origin. Generally, the radial sum of two convex bodies containing the origin is not convex. We show that the radial sum of a star body (with respect to the origin) and any large centered ball is convex, which produces a pair of convex bodies containing the origin whose radial sum is convex.We also investigate the convexity of the intersection body of a convex body containing the origin. Generally, the intersection body of a convex body containing the origin is not convex. Busemann’s theorem states that the intersection body of any centered convex body is convex. We are interested in how to construct convex intersection bodies from convex bodies without any symmetry (especially, central symmetry). We show that the intersection body of the radial sum of a star body (with respect to the origin) and any large centered ball is convex, which produces a convex body with no symmetries whose intersection body is convex.

Détail

DOI : 10.4153/S0008439523000681

Mots-clés : Convexity of the radial sum of convex bodies, Busemann’s theorem, convexity of intersection bodies