ON THE Lp-MINKOWSKI PROBLEM

A volume-normalized formulation of the Lp-Minkowski problem is presented. This formulation has the advantage that a solution is possible for all p ≥ 1, including the degenerate case where the index p is equal to the dimension of the ambient space. A new approach to the Lp-Minkowski problem is presented, which solves the volume-normalized formulation for even data and all p ≥ 1. The Minkowski problem deals with existence, uniqueness, regularity, and stability of closed convex hypersurfaces whose Gauss curvature (as a function of the outer normals) is preassigned. Major contributions to this problem were made by Minkowski [M1], [M2], Aleksandrov [A2], [A3], [A4], Fenchel and Jessen [FJ], Lewy [Le1] [Le2], Nirenberg [N], Calabi [Cal], Pogorelov [P1], [P2], Cheng and Yau [ChY], Caffarelli, Nirenberg, and Spruck [CNS], and others. Variants of the Minkowski problem were presented by Gluck [Gl1] and Singer [Si]. The survey of Gluck [Gl2] still serves as an excellent introduction to the problem. In this article we consider a generalization of the Minkowski problem known as the Lp-Minkowski problem. This generalization was studied in [Lu1] and [LuO]. See Stancu [St1], [St2] and Umanskiy [U] for other recent work on the Lp-Minkowski problem. In [Lu1] a solution to the even Lp-Minkowski problem in R was given for all p ≥ 1 (the case p = 1 is classical), except for p = n. The solution to the even LpMinkowski problem was one of the critical ingredients needed to obtain the sharp affine Lp Sobolev inequality [LuYZ1]. The lack of a solution for the case p = n is troubling. In this article we present a new volume normalized form of the classical Minkowski problem. This problem has a natural Lp analog that can (and will) be solved for all p ≥ 1 for the even data case. It must be emphasized that, except for the critical case p = n, both the Lp-Minkowski problem and the volume normalized Lp-Minkowski problem are equivalent in that a solution to one will quickly and trivially provide a solution to the other. The road to the solution given here to the even volume normalized Lp-Minkowski problem is quite different from the path taken in [Lu1] in solving the even Lp-Minkowski problem. The solution to the volume-normalized even LpMinkowski problem for all p ≥ 1 is needed in [LuYZ2]. A compact convex subset of Euclidean n-space R will be called a convex body. Associated with a convex body K is its support function h(K, · ) : Sn−1 → R which, for u ∈ Sn−1, is defined by h(K,u) = max{u · x : x ∈ K}. For each Received by the editors May 16, 2001 and, in revised form, April 16, 2003. 2000 Mathematics Subject Classification. Primary 52A40. This research was supported, in part, by NSF Grants DMS–9803261 and DMS–0104363. c ©2003 American Mathematical Society