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arXiv · 1406.0811

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

Pedro Freitas·David Krejcirik
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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.

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Pedro Freitas, David Krejcirik. 2014-06-03. Alexandrov's isodiametric conjecture and the cut locus of a surface. https://arxiv.org/abs/1406.0811

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