$C^{1, 1}$ regularity for solutions to the degenerate $L_p$ Dual Minkowski problem

Chen, Li; Tu, Qiang; Wu, Di; Ni, Xiang

doi:10.48550/arxiv.2010.06372

Cited by 1 publication

(1 citation statement)

References 45 publications

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“…Solutions to the Orlicz-Minkowski problem can be found in, e.g., [2,9,26,35,40,45,57,58,59,63,64]. When G = t q /n and ϕ(t) = t p for 0 = p ∈ R, the general dual Orlicz-Minkowski problem reduces to the L p dual Minkowski problem [50]; contributions to this problem can be seen in, e.g., [3,10,11,14,33,38,41,55]. By letting G(u, t) = log t for all (u, t) ∈ S n−1 × (0, ∞), V G (K) for K ∈ K n (o) reduces to the dual entropy of K; in this case one can get the (L p and Orlicz) Aleksandrov problems [1,20,31] (see also [42,68]).…”

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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