Tohoku Mathematical Journal

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

Pedro Freitas and David Krejčiřík

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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.

Article information

Source
Tohoku Math. J. (2) Volume 67, Number 3 (2015), 405-417.

Dates
First available in Project Euclid: 30 November 2015

Permanent link to this document
http://projecteuclid.org/euclid.tmj/1448900034

Digital Object Identifier
doi:10.2748/tmj/1448900034

Mathematical Reviews number (MathSciNet)
MR3430198

Zentralblatt MATH identifier
1341.52012

Subjects
Primary: 53C45: Global surface theory (convex surfaces à la A. D. Aleksandrov)
Secondary: 53A05: Surfaces in Euclidean space 53C22: Geodesics [See also 58E10] 52A15: Convex sets in 3 dimensions (including convex surfaces) [See also 53A05, 53C45] 53A07: Higher-dimensional and -codimensional surfaces in Euclidean n-space

Keywords
Alexandrov's conjecture convex surfaces ellipsoids cut locus symmetrization

Citation

Freitas, Pedro; Krejčiřík, David. Alexandrov's isodiametric conjecture and the cut locus of a surface. Tohoku Math. J. (2) 67 (2015), no. 3, 405--417. doi:10.2748/tmj/1448900034. http://projecteuclid.org/euclid.tmj/1448900034.


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