Curvatures of typical convex bodies— the complete picture

Schneider, Rolf

doi:10.1090/S0002-9939-2014-12245-3

Curvatures of typical convex bodies— the complete picture
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Proc. Amer. Math. Soc. 143 (2015), 387-393 Request permission

Abstract:

It is known that a typical $n$-dimensional convex body, in the Baire category sense, has the property that its set of umbilics of zero curvature has full measure in the boundary of the body. We show that a typical convex body has in addition the following properties. The spherical image of the set of umbilics of zero curvature has measure zero. The set of umbilics of infinite curvature is dense in the boundary and uncountable and its spherical image has full measure in the unit sphere.

References

Karim Adiprasito, Infinite curvature on typical convex surfaces, Geom. Dedicata 159 (2012), 267–275. MR 2944531, DOI 10.1007/s10711-011-9658-0
A. D. Alexandroff, Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it, Leningrad State Univ. Annals [Uchenye Zapiski] Math. Ser. 6 (1939), 3–35 (Russian). MR 0003051
Nicolas Bourbaki, Elements of mathematics. General topology. Part 2, Hermann, Paris; Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966. MR 0205211
Herbert Busemann, Convex surfaces, Interscience Tracts in Pure and Applied Mathematics, no. 6, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. MR 0105155
Peter M. Gruber, Die meisten konvexen Körper sind glatt, aber nicht zu glatt, Math. Ann. 229 (1977), no. 3, 259–266 (German). MR 442825, DOI 10.1007/BF01391471
Peter M. Gruber, Results of Baire category type in convexity, Discrete geometry and convexity (New York, 1982) Ann. New York Acad. Sci., vol. 440, New York Acad. Sci., New York, 1985, pp. 163–169. MR 809203, DOI 10.1111/j.1749-6632.1985.tb14550.x
Peter M. Gruber, Baire categories in convexity, Handbook of convex geometry, Vol. A, B, North-Holland, Amsterdam, 1993, pp. 1327–1346. MR 1243011
Victor Klee, Some new results on smoothness and rotundity in normed linear spaces, Math. Ann. 139 (1959), 51–63 (1959). MR 115076, DOI 10.1007/BF01459822
Rolf Schneider, On the curvatures of convex bodies, Math. Ann. 240 (1979), no. 2, 177–181. MR 524665, DOI 10.1007/BF01364632
Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. MR 1216521, DOI 10.1017/CBO9780511526282
Rolf Schneider and Wolfgang Weil, Stochastic and integral geometry, Probability and its Applications (New York), Springer-Verlag, Berlin, 2008. MR 2455326, DOI 10.1007/978-3-540-78859-1
Tudor Zamfirescu, The curvature of most convex surfaces vanishes almost everywhere, Math. Z. 174 (1980), no. 2, 135–139. MR 592910, DOI 10.1007/BF01293533
Tudor Zamfirescu, Nonexistence of curvature in most points of most convex surfaces, Math. Ann. 252 (1980), no. 3, 217–219. MR 593634, DOI 10.1007/BF01420084
Tudor Zamfirescu, Using Baire categories in geometry, Rend. Sem. Mat. Univ. Politec. Torino 43 (1985), no. 1, 67–88. MR 859850
Tudor Zamfirescu, Curvature properties of typical convex surfaces, Pacific J. Math. 131 (1988), no. 1, 191–207. MR 917873
Tudor Zamfirescu, Baire categories in convexity, Atti Sem. Mat. Fis. Univ. Modena 39 (1991), no. 1, 139–164. MR 1111764
Tudor Zamfirescu, The majority in convexity, Editura Univ. Bucureşti, Bucharest, 2009.