Continuity of the solution to the dual Minkowski problem for negative indices

“…Regarding the q-th dual curvature measures is the remarkable dual Minknowski problem [30]: given a real number q and a nonzero finite Borel measure µ defined on S n−1 , can one find a K ∈ K n (o) so that µ = C q (K, •), with C q (K, •) the q-th dual curvature measure of K ∈ K n (o) ? Since its introduction, the dual Minkowski problem has received a lot of attention, see e.g., [5,7,16,25,28,39,42,61,62,66,67].…”

Section: Introductionmentioning

confidence: 99%

On the Musielak-Orlicz-Gauss image problem

Huang,

Xing,

et al. 2021

Preprint

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In the present paper we initiate the study of the Musielak-Orlicz-Brunn-Minkowski theory for convex bodies. In particular, we develop the Musielak-Orlicz-Gauss image problem aiming to characterize the Musielak-Orlicz-Gauss image measure of convex bodies. For a convex body K, its Musielak-Orlicz-Gauss image measure, denoted by C Θ (K, •), involves a triple Θ = (G, Ψ, λ) where G and Ψ are two Musielak-Orlicz functions defined on S n−1 × (0, ∞) and λ is a nonzero finite Lebesgue measure on the unit sphere S n−1 . Such a measure can be produced by a variational formula of V G,λ (K) (the general dual volume of K with respect to λ) under the perturbations of K by the Musielak-Orlicz addition defined via the function Ψ. The Musielak-Orlicz-Gauss image problem contains many intensively studied Minkowski type problems and the recent Gauss image problem as its special cases. Under the condition that G is decreasing on its second variable, the existence of solutions to this problem is established.

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“…Since the groundbreaking work of Huang, Lutwak, Yang and Zhang, the intrinsic partial differential equations that arise within the dual Brunn-Minkowski theory which wait a full 40 years after the birth of the dual theory to emerge. Some significant breakthroughs of dual Minkowski problems have been achieved [6,8,11,31,36,37,41,52,65,66,70,71,76] but many cases remain open.…”

Section: Introductionmentioning

confidence: 99%