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Christoffel's problem for general convex bodies

  • William J. Firey (a1)

Christoffel's problem, in its classical form, asks for the determination of necessary and sufficient conditions on a function φ, defined over the unit spherical surface Ω, in order that there exist a convex body K for which φ (u) is the sum of the principal radii of curvature at that boundary point of K where the outer unit normal is u. The figures Ω and K are in Euclidean n-dimensional space (n ≥ 3). It is assumed that φ is continuously differentiable and that K is of sufficient smoothness. A solution of Christoffe's problem was given in [6]. Yet that treatment is rather unsatisfactory in that the smoothness restrictions are set by the method rather than the problem, cf. [5; p. 60]. The present paper overcomes this defect. To do this it is first necessary to generalize the original problem so as to seek conditions on a measure M, defined over the Borel sets of Ω, in order that M be a first order area function for a convex body K. When K has sufficient smoothness, then φ is the Radon-Nikodym derivative of M with respect to surface area measure on Ω. It is this generalized Christoffel problem which is solved in what follows.

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1.Alexandrov, A. D., “Zur Theorie der gemischten Volumina von konvexen Körper, I. Verailgemeinering einiger Begriffe der Theorie der konvexen köorper”, Mat. Sbornik, N. S., 2 (1937), 947972. (Russian with German summary.)
2.Alexandrov, A. D., “Zur Theorie der gemischten Volumina von konvexen Körper, III. Die Erweiterung zweier Lehrsätze Minkowskis über die konvexen Polyeder auf beliebige konvexe Flächen”, Mat. Sbornik, N. S., 3 (1938), 2746. (Russian with German summary.)
3.Bonnesen, T. and Fenchel, W., Theorie der konvexen Körper (Berlin, 1934).
4.Busemann, H., Convex Surfaces (New York, 1958).
5.Fenchel, W. and Jessen, B., “Mengenfunktionen und konvexe Korper”, Det Kgl. Danske Videnskab. Selskab, Math.-fys. Medd., 16 (1938), 3.
6.Firey, W. J., “The Determination of Convex Bodies from Their Mean Radius of Curvature Functions”, Mathematika, 14, 2 (1967), 113.
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  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
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