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Nonexistence of curvature in most points of most convex surfaces. (English) Zbl 0427.53002


MSC:

53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
52A10 Convex sets in \(2\) dimensions (including convex curves)
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
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References:

[1] Aleksandrov, A.D.: Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it (Russian). Uchenye Zapiski Leningrad. Gos. Univ. Math. Ser.6, 3-35 (1939)
[2] Busemann, H.: Convex Surfaces. New York: Interscience Publishers 1958 · Zbl 0196.55101
[3] Busemann, H., Feller, W.: Krümmungseigenschaften konvexer Flächen. Acta Math.66, 1-47 (1935) · Zbl 0012.27404 · doi:10.1007/BF02546515
[4] Gruber, P.M.: Die meisten konvexen Körper sind glatt, aber nicht zu glatt. Math. Ann.229, 259-266 (1977) · Zbl 0349.52004 · doi:10.1007/BF01391471
[5] Klee, V.: Some new results on smoothness and rotundity in normed linear spaces. Math. Ann.139, 51-63 (1959) · Zbl 0092.11602 · doi:10.1007/BF01459822
[6] Schneider, R.: On the curvatures of convex bodies. Math. Ann.240, 177-181 (1979) · Zbl 0392.52005 · doi:10.1007/BF01364632
[7] Zamfirescu, T.: The curvature of most convex surfaces vanishes almost everywhere (to appear in Math. Z.) · Zbl 0423.53003
[8] Zamfirescu, T.: Intersections of tangent convex curves (to appear) · Zbl 0486.52002
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