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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Extreme points of the distance function on convex surfaces
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by Tudor Zamfirescu PDF
Trans. Amer. Math. Soc. 350 (1998), 1395-1406 Request permission

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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Additional Information
  • Tudor Zamfirescu
  • Affiliation: Fachbereich Mathematik, Universität Dortmund, 44221 Dortmund, Germany
  • Email: tudor.zamfirescu@mathematik.uni-dortmund.de
  • Received by editor(s): April 17, 1996
  • © Copyright 1998 American Mathematical Society
  • 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