A little over a hundred years ago E. B. Christoffel in  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 , , 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 , D. Hilbert , T. Kubota , J. Favard , and W. Suss . A succinct discussion of the question is in Busemann  but the footnote on p. 68, intended to rectify the discussion in Bonnesen and Fenchel , is also not correct. A sufficient, but not necessary condition was found by A. V. Pogorelov in .
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