Alexandrov's isodiametric conjecture and the cut locus of a surface

TL;DR

This paper proves Alexandrov's isodiametric conjecture for convex ellipsoid surfaces and extends it to higher dimensions for spherically symmetric convex manifolds, using a novel symmetrization method.

Contribution

It establishes the conjecture for ellipsoids and higher-dimensional spherically symmetric manifolds, introducing a new symmetrization technique.

Findings

Conjecture holds for convex ellipsoid surfaces.

Extension of the conjecture to higher dimensions.

New symmetrization procedure developed.

Abstract

We prove that Alexandrov's conjecture relating the area and diameter of a convex surface holds for the surface of a general ellipsoid. This is a direct consequence of a more general result which estimates the deviation from the optimal conjectured bound in terms of the length of the cut locus of a point on the surface. We also prove that the natural extension of the conjecture to general dimension holds among closed convex spherically symmetric Riemannian manifolds. Our results are based on a new symmetrization procedure which we believe to be interesting in its own right.