Christoffel's problem for general convex bodies

Extract

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.