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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
DOI: https://doi.org/10.1090/S0002-9947-98-02106-0
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
Email: tudor.zamfirescu@mathematik.uni-dortmund.de

Received by editor(s): April 17, 1996
Article copyright: © Copyright 1998 American Mathematical Society