On the Local Convexity of Intersection Bodies of Revolution

Abstract

Abstract. One of the fundamental results in Convex Geometry is Busemann's theorem, which states that the intersection body of a symmetric convex body is convex. Thus, it is only natural to ask if there is a quantitative version of Busemann's theorem, i.e., if the intersection body operation actually improves convexity. In this paper we concentrate on the symmetric bodies of revolution to provide several results on the (strict) improvement of convexity under the intersection body operation. It is shown that the intersection body of a symmetric convex body of revolution has the same asymptotic behavior near the equator as the Euclidean ball. We apply this result to show that in sufficiently high dimension the double intersection body of a symmetric convex body of revolution is very close to an ellipsoid in the Banach-Mazur distance. We also prove results on the local convexity at the equator of intersection bodies in the class of star bodies of revolution.

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

“…We emphasize that central symmetry essentially works for Busemann’s theorem and its generalization. In [1], the local convexity of intersection bodies of symmetric convex bodies of revolution was investigated. Namely, it is still open to concretely give convex bodies with no symmetries whose intersection bodies are globally convex .…”