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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Points joined by three shortest paths on convex surfaces

Author: Tudor Zamfirescu
Journal: Proc. Amer. Math. Soc. 123 (1995), 3513-3518
MSC: Primary 52A15; Secondary 53C45
MathSciNet review: 1273530
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Abstract: Let S be a convex surface and $ x \in S$. It is shown here that the set of all points of S joined with x by at least three shortest paths can be dense in S. It is proven that, in fact, in the sense of Baire categories most convex surfaces have this property, for any x. Moreover, on most convex surfaces, for most of their points, there is just one farthest point (in the intrinsic metric), and precisely three shortest paths lead to that point.

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Keywords: Baire categories, generic convex surfaces, geodesic segments
Article copyright: © Copyright 1995 American Mathematical Society

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