The L dual Minkowski problem for p > 1 and q > 0

Abstract

General L p dual curvature measures have recently been introduced by Lutwak, Yang and Zhang [24]. These new measures unify several other geometric measures of the Brunn-Minkowski theory and the dual Brunn-Minkowski theory. L p dual curvature measures arise from qth dual inrinsic volumes by means of Alexandrov-type variational formulas. Lutwak, Yang and Zhang [24] formulated the L p dual Minkowski problem, which concerns the characterization of L p dual curvature measures.In this paper, we solve the existence part of the L p dual Minkowski problem for p > 1 and q > 0, and we also discuss the regularity of the solution.

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

“…where h K denotes the support function of K (see Section 2 for notation and most definitions). Again, this result recovers (in a slightly different form) and strengthens the solutions to the Orlicz-Minkowski problem in [18,Theorem 1.2] and the L p dual Minkowski problem in [1,Theorem 1.2]. In Theorem 6.4, we use the same approximation technique to prove a variant of [12,Theorem 6.4] in the case when G t < 0.…”

“…When ψ(t) = t p for p > 1 and G(t, u) = t q φ(u) for q > 0 and φ ∈ C + (S n−1 ), Theorem 4.4 recovers the solution to the L p dual Minkowski problem for discrete measures by Böröczky and Fodor [1, Theorem 1.1]. The techniques in these works are similar and based on those in [23], but some of our arguments differ from and are rather more complicated than those in [1,18,25]. In particular, the general volume V G (·) prohibits the use of Minkowski's inequality as in [18,25], and in general the two-variable function G, and the lack of homogeneity of G and ψ, require a somewhat more delicate analysis than the special case considered in [1].…”

“…In particular, the general volume V G (·) prohibits the use of Minkowski's inequality as in [18,25], and in general the two-variable function G, and the lack of homogeneity of G and ψ, require a somewhat more delicate analysis than the special case considered in [1]. On the other hand, we are able to avoid some constructions in [1] by making use of the absolute continuity of C G,ψ (K, ·) with respect to the surface area measure of K proved in Proposition 5.2(ii).…”

On the general dual Orlicz-Minkowski problem

“…Note that the general dual volume V G (·) was used to derive the general dual Orlicz curvature measures and hence plays central roles in establishing the existence of solutions to the recently proposed general dual Orlicz-Minkowski problem [13,15]. When G(t, u) = 1 n t n , one gets V G (K) = V (K), and when G(t, u) = 1 n t q for q = 0, n, V G (K) becomes the qth dual volume V q (K) which plays fundamental roles in the dual Brunn-Minkowski theory [35,36,37] and the L p dual Minkowski problem (see e.g., [2,4,6,7,24,25,42,61]). When G(t, u) = G(t, e 1 ) for all (t, u) ∈ (0, ∞) × S n−1 , V G (K) becomes the dual Orlicz-quermassintegral in [63]; while if G(t, u) = t 0 φ(ru)r n−1 dr or G(t, u) = ∞ t φ(ru)r n−1 dr for some function φ : R n → (0, ∞), then V G (K) becomes the general dual Orlicz quermassintegral in [55].…”

“…It has been proved in [34, Propositions 3.1 and 3.3] that the solutions to the polar Orlicz-Minkowski problem for discrete measures must be polytopes, the convex hulls of finite points in R n . It is well-known that all convex bodies can be approximated by polytopes, and hence to study the Minkowski type problems for discrete measures is very important and receives extensive attention, see e.g., [2,3,11,15,21,23,26,29,30,53,65,66,67].…”