Infinite curvature on typical convex surfaces
The main result of the paper settles a long-standing open problem of Zamfirescu on the curvature on typical (in the sense of Baire's first category) convex surfaces. It was already shown by \textit{T. Zamfirescu} [Math. Z. 174, 135--139 (1980; Zbl 0423.53003)] that a typical convex body has points of curvature 0 in all tangent directions almost everywhere, and typical convex bodies in the plane contain uncountable many boundary points of infinite curvature [Pac. J. Math. 131, No. 1, 191--207 (1988; Zbl 0637.52005)]. Here the author shows that in any dimension, a typical convex surface contains at least one point of infinite curvature in all tangent directions.
scientific article; zbMATH DE number 6062586