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

Local behaviour of area functions of convex bodies

“…By letting a tend to zero, we see that (3) is a consequence of (1). Necessary and sufficient conditions for Φ to be an area function of order one are given in [4] and [5]. Inequality (1) for p = 1 was proved in the latter paper and plays a significant part.…”

“…Since K z £ iΓ 4 and S n^( K 9 Ω) is increasing in ϋΓ, it follows that (7) holds for j = 3, p = n -1. For the cases 1 ^ p < n -1 a more elaborate argument is needed.…”

A Fourier transform approach to Christoffel’s problem

“…It is named in recognition of Christoffel's work [7], however Alexandrov [1] and [3] pointed to gaps in his work. As Firey [9] points out, these gaps arose since the conditions derived by Christoffel were not sufficient to guarantee that the resultant surface was a closed convex surface. Busemann [5] provides a description of the problem and a number of unsuccessful attempts, though even here there are also some mistakes.…”

“…Busemann [5] provides a description of the problem and a number of unsuccessful attempts, though even here there are also some mistakes. The problem, as described above, was eventually solved independently by Firey [9], [10] and Berg [4]. The solutions provided by Berg and Firey do not yield a characterization which is easy to apply to specific problems.…”

“…Firey's solution [9] to the smooth case of Christoffel's problem uses the Green's function method to solve a Poisson equation for the position function of the boundary surface of K. He then shows that the positivity of a certain integral transform is the characteristic property for a given function to be the mean radius of curvature of a closed convex surface.…”