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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


Extreme points of the distance function
on convex surfaces

Author: Tudor Zamfirescu
Journal: Trans. Amer. Math. Soc. 350 (1998), 1395-1406
MSC (1991): Primary 52A15, 53C45
MathSciNet review: 1458314
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Abstract: We first see that, in the sense of Baire categories, many convex surfaces have quite large cut loci and infinitely many relative maxima of the distance function from a point. Then we find that, on any convex surface, all these extreme points lie on a single subtree of the cut locus, with at most three endpoints. Finally, we confirm (both in the sense of measure and in the sense of Baire categories) Steinhaus' conjecture that ``almost all" points admit a single farthest point on the surface.

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Tudor Zamfirescu
Affiliation: Fachbereich Mathematik, Universität Dortmund, 44221 Dortmund, Germany

PII: S 0002-9947(98)02106-0
Received by editor(s): April 17, 1996
Article copyright: © Copyright 1998 American Mathematical Society