On the Christoffel-minkowski Problem of Firey’s P-sum

The classical Brunn-Minkowski theory for convex bodies was developed from a few basic concepts: support functions, Minkowski combinations, and mixed volumes. As a special case of mixed volumes, the Quermassintegrals are important geometrical quantities of a convex body, and surface area measures are local versions of Quermassintegrals. The Christoffel-Minkowski problem concerns with the existence of convex bodies with prescribed surface area measure, for details please refer to [17, 3, 8, 18]. In 1962, Firey [5] generalized the Minkowski combination to p-sums from p = 1 to p ≥ 1. Later, Lutwak [13, 14] showed that Firey’s p-sum also leads to a Brunn-Minkowski theory for each p > 1. This theory has found many geometry applications, see for example, [16] and its references. It was also shown in [13] that the classical surface area measures could be extended to the p-sum case. So it is natural to consider a generalization of the classical Christoffel-Minkowski problem for each p > 1. The generalized Minkowski problem has been treated in [13, 15, 7, 4]. In this paper we study the remaining case, which may be called the Christoffel-Minkowski problem of p-sum. First we introduce some notations and relevant results. Let Kn+1 denote the set of convex bodies (compact, convex subsets with nonempty interiors) in Euclidean space Rn+1, and let Kn+1 0 denote the set of convex bodies containing the origin in their interiors. For K ∈ Kn+1, let hK = h(K, ·) : Sn → R denote the support function of K, and let W0(K),W1(K), ..., Wn+1(K) denote the Quermassintegrals of K (see, for example [18]). Thus W0(K) = V (K), the volume of K and Wn+1 = V (B) = ωn+1, where B is the unit ball in Rn+1. For each Firey p-sum, Lutwak [13] defined the mixed p-Quermassintegrals by n + 1− k p Wp,k(K, L) = lim 2→0+ Wk(K +p 2 · L)−Wk(K) 2 . (1.1)