General volumes in the Orlicz–Brunn–Minkowski theory and a related Minkowski problem II

Abstract

The general dual volume V G (K) and the general dual Orlicz curvature measure C G,ψ (K, ·) were recently introduced for functions G : (0, ∞) × S n−1 → (0, ∞) and convex bodies K in R n containing the origin in their interiors. We extend V G (K) and C G,ψ (K, ·) to more general functions G : [0, ∞) × S n−1 → [0, ∞) and to compact convex sets K containing the origin (but not necessarily in their interiors). Some basic properties of the general dual volume and of the dual Orlicz curvature measure, such as the continuous dependence on the underlying set, are provided. These are required to study a Minkowski-type problem for the dual Orlicz curvature measure. We mainly focus on the case when G and ψ are both increasing, thus complementing our previous work.The Minkowski problem asks to characterize Borel measures µ on S n−1 for which there is a convex body K in R n containing the origin such that µ equals C G,ψ (K, ·), up to a constant. A major step in the analysis concerns discrete measures µ, for which we prove the existence of convex polytopes containing the origin in their interiors solving the Minkowski problem. For general (not necessarily discrete) measures µ, we use an approximation argument. This approach is also applied to the case where G is decreasing and ψ is increasing, and hence augments our previous work. When the measures µ are even, solutions that are originsymmetric convex bodies are also provided under some mild conditions on G and ψ. Our results generalize several previous works and provide more precise information about the solutions of the Minkowski problem when µ is discrete or even.2 RICHARD J. GARDNER, DANIEL HUG, SUDAN XING, AND DEPING YE problem; see e.g. [23,30,32]. The case p = 0 is of particular significance because the L 0 surface area measure, the so-called cone volume measure, is an affine invariant. The L 0 or logarithmic Minkowski problem is challenging and was only solved recently for even measures by Böröczky, Lutwak, Yang, and Zhang [5]. More recent contributions to the logarithmic Minkowski problem are [3,42] and further references and background on the L p Minkowski problem may be found in [20,36].Recent seminal work of Huang, Lutwak, Yang, and Zhang [20] brought new ingredients, the qth dual curvature measures, to the family of Minkowski problems. These measures are obtained via a first-order variation of the qth dual volume with respect to L 0 addition of convex bodies (see [20, Theorem 4.5]), the case q = n being the L 0 surface area. The authors of [20] posed a corresponding Minkowski problem-the dual Minkowski problem-of finding necessary and/or sufficient conditions for a measure µ on S n−1 to be the qth dual curvature measure of some convex body, and they provided a partial solution when µ is even. Clearly, the logarithmic Minkowski problem is a special case of the dual Minkowski problem. Naturally, the dual Minkowski problem has become important for the dual Brunn-Minkowski theory introduced by Lutwak [28,29]. Since [20], progress includes a complete solution for q < 0...

A flow method for the dual Orlicz–Minkowski problem

“…(1), which contained Theorem A (1) as a special case, see [21,Theorem 6.4]. In the sequel [23], they obtained another existence result about Eq.…”

“…The so called L p dual curvature measures partially unified various measures in the L p Brunn-Minkowski theory and the dual Brunn-Minkowski theory. Very recently, the L p dual curvature measures were developed to the Orlicz case [21,23], which unified more curvature measures and were named dual Orlicz curvature measures. These curvature measures are of central importance in the dual Orlicz-Brunn-Minkowski theory, and the corresponding Minkowski problems are called the dual Orlicz-Minkowski problems.…”

“…See the next section for more details. For convenience, we here have used a slightly different notation in the definition of C ϕ,G (K, ·) from that of [21,23]. The dual Orlicz-Minkowski problem asks what are the necessary and sufficient conditions for a Borel measure µ on the unit sphere S n−1 to be a multiple of the dual Orlicz curvature measure of a convex body K. Namely, this problem is to find a convex body K ⊂ R n such that (1) c d C ϕ,G (K, ·) = dµ on S n−1 for some positive constant c. When the Radon-Nikodym derivative of µ with respect to the spherical measure on S n−1 exists, namely dµ = 1 n f dx for a non-negative integrable function f , the equation (1) is reduced into…”

“…(1) or Eq. (2) is not known and can not be set to 1 in general, see [21,23]. The equation of the Orlicz-Minkowski problem also contains a constant c which is not equal to 1 in general, see [26,30,38].…”

“…(2). We may need to impose strong restrictions on ϕ and G to obtain the uniqueness, see our discussion Theorem 3 for G(y) = G(|y|) and see also [23,Theorem 6.5]. Due to the lack of uniqueness of Eq.…”