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Journal of Convex Analysis 19 (2012), No. 2, 385--391
Copyright Heldermann Verlag 2012

Large Curvature on Typical Convex Surfaces

Karim Adiprasito
Institut für Mathematik, Freie Universität, Arnimallee 2, 14195 Berlin, Germany

Tudor Zamfirescu
Fakultät für Mathematik, Technische Universität, Vogelpothsweg 87, 44227 Dortmund, Germany

We show in this paper that on most convex surfaces there exist points with arbitrarily large lower curvature in every tangent direction. Moreover, we show that, astonishingly, on most convex surfaces, although the set of points with curvature 0 in every tangent direction has full measure, it contains no pair of opposite points, i.e. points admitting parallel supporting planes.

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