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Critical Points on Convex Surfaces

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Abstract

The cut locus C(x) of some point x on an open convex surface is a forest and has measure zero. However, we show here that topologically it can be quite large, namely residual. All critical points with respect to x belong to C(x). We also show that, irrespective of how large C(x) might be, there is a Jordan arc in C(x) containing all critical points.

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Authors and Affiliations

  1. Fachbereich Mathematik, Universität Dortmund, 44221, Dortmund, Germany

    Tudor Zamfirescu

  2. Institute of Mathematics “Simion Stoilow”, Roumanian Academy, Bucharest, Roumania

    Tudor Zamfirescu

  3. College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, 050024, People’s Republic of China

    Tudor Zamfirescu

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  1. Tudor Zamfirescu
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Correspondence to Tudor Zamfirescu.

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Zamfirescu, T. Critical Points on Convex Surfaces. Results Math 73, 19 (2018). https://doi.org/10.1007/s00025-018-0777-x

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  • DOI: https://doi.org/10.1007/s00025-018-0777-x

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