Extreme points of the distance function on convex surfaces

Zamfirescu, Tudor

doi:10.1090/S0002-9947-98-02106-0

Extreme points of the distance function on convex surfaces
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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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