On the Musielak-Orlicz-Gauss image problem

Abstract

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.

A flow approach to the Musielak-Orlicz-Gauss image problem

“…There is a growing body work to these problems and we only name a few [3,4,5,10,16,17,18,21,30,34,35,41,44,57,58]. The list is far from complete and more can be found in the references in, e.g., [31,32,33,60]. On the one hand, these Minkowski type problems can be reformulated by Monge-Ampère type equations (assume enough smoothness) and hence largely enrich the theory of fully nonlinear PDEs.…”

“…al. [31], where they also introduced the Musielak-Orlicz-Gauss image problem aiming to characterize the Musielak-Orlicz-Gauss image measure C Θ (Ω, •) for convex body Ω in R n+1 . Here, Θ = (G, Ψ, λ) is a given triple with two Musielak-Orlicz functions G and Ψ defined on (0, +∞) × S n , and λ a nonzero finite Borel measure on S n .…”

“…Here, Θ = (G, Ψ, λ) is a given triple with two Musielak-Orlicz functions G and Ψ defined on (0, +∞) × S n , and λ a nonzero finite Borel measure on S n . Under the condition that G is decreasing on its first variable, the existence of solutions to the Musielak-Orlicz-Gauss image problem is established in [31]. See [54] for a special case, the L p Gauss image problem and its solutions.…”

“…The flow technique has been proved to be effective and powerful in solving the Minkowski type and Gauss image problems [7,8,13,14,15,20,37,39,40,53]. The idea behind the flow technique is the fact that the Minkowski type and Gauss image problems can be reformulated as a Monge-Ampère type equation on S n , and this indeed works for the (extended) Musielak-Orlicz-Gauss image problem due to equation (1.11) (see [31]).…”

“…By dξ, we mean the usual spherical measure on S n . The Musielak-Orlicz-Gauss image measure of Ω ∈ K 0 was proposed in [31] as follows.…”