Tohoku Mathematical Journal
2015
September
SECOND SERIES VOL. 67, NO. 3

Tohoku Math. J.
67 (2015), 405-417

Title ALEXANDROV'S ISODIAMETRIC CONJECTURE AND THE CUT LOCUS OF A SURFACE

Author Pedro Freitas and David Krejčiřík

(Received January 27, 2014, revised May 28, 2014)
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.

Mathematics Subject Classification. Primary 53C45; Secondary 53A05, 53C22, 52A15, 53A07.
Key words and phrases. Alexandrov's conjecture, convex surfaces, ellipsoids, cut locus, symmetrization.

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