Area and antipodal distance in convex hypersurfaces

TL;DR

This paper derives lower bounds for the surface area and mean width of convex hypersurfaces, extending known volume bounds for Riemannian spheres across all dimensions.

Contribution

It introduces new lower bounds for surface area and mean width of convex hypersurfaces, generalizing previous results on sphere volume bounds.

Findings

Established a lower bound for surface area based on displacement under continuous maps.

Proved a sharp lower bound for the mean width of convex hypersurfaces.

Extended the validity of a volume bound for Riemannian spheres to all dimensions for convex hypersurfaces.

Abstract

We establish a lower bound for the surface area of a closed, convex hypersurface in Euclidean space in terms of its displacement under continuous maps. As a result, a hypothesized lower bound for the volume of a Riemannian $n$-sphere, proved by Berger in dimension $n=2$ and disproved by Croke in dimensions $n \geq 3$, is valid for convex hypersurfaces in all dimensions. We also establish a sharp lower bound for the mean width of a convex hypersurface.