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The determination of convex bodies from their mean radius of curvature functions

  • W. J. Firey (a1)
Extract

A little over a hundred years ago E. B. Christoffel in [6] asserted a proposition concerning the determination of a surface in Euclidean 3-space from a specification of its mean radius of curvature as a function of the outer normal direction. In that paper an assumption was made which limited the class of surfaces considered to be boundaries of convex bodies. The argument rested on the construction of functions describing the co-ordinates of surface points corresponding to outer normal directions as solutions of certain partial differential equations involving the mean radius of curvature. However, it was pointed out by A. D. Alexandrov [1], [2], that the conditions laid down by Christoffel on the preassigned mean radius of curvature function were not sufficient to ensure that that function actually be a mean radius of curvature function of a closed convex surface. Hence Christofrel's discussion is incomplete. Different and similarly incomplete treatments of Christoffelés problem were given by A. Hurwitz [9], D. Hilbert [8], T. Kubota [11], J. Favard [7], and W. Suss [13]. A succinct discussion of the question is in Busemann [5] but the footnote on p. 68, intended to rectify the discussion in Bonnesen and Fenchel [4], is also not correct. A sufficient, but not necessary condition was found by A. V. Pogorelov in [12].

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1.Alexandrov, A. D., “Über die Frage nach der Existenz eines konvexen Körpers bei dem die Summe der Hauptkrümmungsradien eine gegebene positive Funktion ist, welche die Bedingungen der Geschlossenheit genügt.” Doklady Akad. Nauk, 14 (1937), 5960.
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.Blaschke, W., Kreis undKugel (Leipzig, 1916).
4.Bonnesen, T. and Fenchel, W., Theorie der konvexen Körper (Berlin, 1934).
5.Busemann, H., Convex surfaces (New York, 1958).
6.Christoffel, E. B., “Ueber die Bestimmung der Gestalt einer krummen Oberfläche durch lokale Messungen auf derselben,” J. für die reine und angew. Math., 64 (1865), 193-209 = Werke, vol. I (Leipzig and Berlin, 1910, 162177).
7.Favard, J., “Sur la determination des surfaces convexes,” Acad. royale Belgique, Bull. cl. Sciences (5), 19 (1933), 6575.
8.Hilbert, D., Grundzüge einer allgemeinen Theorie der linearen Integralgieichimgen (Leipzig and Berlin, 1912).
9.Hurwitz, A., “Sur quelques applications géométriques des séries de Fourier,” Ann. École norm!. (3), 13 (1902), 357408.
10.Kellogg, O., Foundations of potential theory (New York, 1929).
11.Kubota, T., “Über die Eibereiche im n–dimensionalen Raum,” Sci. Rep. Tôhoku Univ., 14 (1925), 399402.
12.Pogorelov, A. V., “On the question of the existence of a convex surface with a given sum of the principal radii of curvature,” Uspekhi Mat. Nauk, 8 (1953), 127130 (Russian).
13.Suss, W., “Bestimmung einer geschlossenen konvexen Fläche durch die Summe ihrer Hauptkrümmungsradien,” Math. Annalen, 108 (1933), 143148.
14.Vincensini, P., “Sur le prolongement des series lineaires de corps convexes. Applications,” Rendiconti del Circ. Matem. di Palermo, 60 (1936), 361372.
15.Vincensini, P., Corps convexes. Séries linéaires. Domaines vectoriels. Mem. des Sci. Math., 44 (Paris, 1938).
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Mathematika
  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
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