Necessary subspace concentration conditions for the even dual Minkowski problem

Abstract

Abstract. We prove tight subspace concentration inequalities for the dual curvature measures Cq(K, ·) of an n-dimensional origin-symmetric convex body for q ≥ n + 1. This supplements former results obtained in the range q ≤ n.

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

“…Note that if dim K < n, then C G,ψ (K, E) = 0, since S n−1 ∩ N(K, o) * is then at most (n − 2)-dimensional. Furthermore, if dim K = n, then in view of (8) and (15), the integral in (58) may equivalently be taken over α α α *…”

“…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 by Zhao [38], solutions for even µ in [4,6,15,39], and solutions via curvature flows and partial differential equations in [8,24,26].An important extension of the dual Minkowski problem was carried out by Lutwak, Yang, and Zhang [33], who introduced L p dual curvature measures and posed a corresponding L p dual Minkowski problem. In [33], the L 0 addition in [20] is replaced by L p addition, while the qth dual volume remains unchanged.…”

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

“…The case of the L p dual Minkowski problem for even measures has received much attention but is not discussed here, see Böröczky, Lutwak, Yang, Zhang [5] concerning the L p surface area S p (K, ·), Böröczky, Lutwak, Yang, Zhang, Zhao [6], Jiang Wu [20] and Henk, Pollehn [15] concerning the qth dual curvature measure C q (K, ·), and Huang, Zhao [19] concerning the L p dual curvature measure for detailed discussion of history and recent results.…”

“…For a given non-trivial finite Borel measure µ, find a convex body K ∈ K n o such that (14) d C q (K, Q, ·) = h p K dµ. It is natural to assume that H n−1 (Ξ K ) = 0 in (14) for (15) Ξ K = {x ∈ ∂K : there exists exterior normal u ∈ S n−1 at x with h K (u) = 0}, which property ensures that the surface area measure S(K, ·) is absolute continuous with respect to C q (K, Q, ·) (see Corollary 6.2). Actually, if q = n and Q = B n , then d C n (K, ·) = 1 n h K dS(K, ·), and [11] and [16] consider the problem (16) dS(K, ·) = nh p−1 K dµ, where the results of [16] about (16) yield the uniqueness of the solution in (16) for q = n, p > 1 and Q = B n only under the condition H n−1 (Ξ K ) = 0 (see Section 4 for more detailed discussion).…”

The 𝐿_{𝑝} Aleksandrov problem for origin-symmetric polytopes

“…The dual Minkowski problem miraculously contains problems such as the Aleksandrov problem (q = 0) and the logarithmic Minkowski problem (q = n) as special cases. The problem quickly became the center of attention, see, for example, [6,9,13,19,22,25,26,30,40,49,50].…”