## 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.## References

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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