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

Abstract

The general volume of a star body, a notion that includes the usual volume, the qth dual volumes, and many previous types of dual mixed volumes, is introduced. A corresponding new general dual Orlicz curvature measure is defined that specializes to the (p, q)-dual curvature measures introduced recently by Lutwak, Yang, and Zhang. General variational formulas are established for the general volume of two types of Orlicz linear combinations. One of these is applied to the Minkowski problem for the new general dual Orlicz curvature measure, giving in particular a solution to the Minkowski problem posed by Lutwak, Yang, and Zhang for the (p, q)-dual curvature measures when p > 0 and q < 0. A dual Orlicz-Brunn-Minkowski inequality for general volumes is obtained, as well as dual Orlicz-Minkowski-type inequalities and uniqueness results for star bodies. Finally, a very general Minkowski-type inequality, involving two Orlicz functions, two convex bodies, and a star body, is proved, that includes as special cases several others in the literature, in particular one due to Lutwak, Yang, and Zhang for the (p, q)-mixed volume.

A flow method for the dual Orlicz–Minkowski problem

“…See the next section for more details. For convenience, we here have used a slightly different notation in the definition of C ϕ,G (K, ·) from that of [21,23]. The dual Orlicz-Minkowski problem asks what are the necessary and sufficient conditions for a Borel measure µ on the unit sphere S n−1 to be a multiple of the dual Orlicz curvature measure of a convex body K. Namely, this problem is to find a convex body K ⊂ R n such that (1) c d C ϕ,G (K, ·) = dµ on S n−1 for some positive constant c. When the Radon-Nikodym derivative of µ with respect to the spherical measure on S n−1 exists, namely dµ = 1 n f dx for a non-negative integrable function f , the equation (1) is reduced into…”

On the general dual Orlicz-Minkowski problem

“…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]. See [13] for more special cases. It has been proved that V G (K i ) → V G (K) for G : (0, ∞) × S n−1 → (0, ∞) being continuous and…”

“…It is easy to check that V G (·) in general is not homogeneous on K n o and/or K n (o) . Note that the general dual volume V G (·) can be defined not only for convex bodies, but also for star-shaped sets, see [13] for more details.…”

“…Let G : (0, ∞) × S n−1 → (0, ∞) be a continuous function. In [13], the general volume of a convex body K ∈ K n (o) , denoted by V G (K), is proposed to be…”

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This paper gives a systematic study to the general dual-polar Orlicz-Minkowski problem (e.g., Problem 4.1). This problem involves the general dual volume V G (·) recently proposed in [13,15] in order to study the general dual Orlicz-Minkowski problem. As V G (·) extends the volume and the qth dual volume, the general dual-polar Orlicz-Minkowski problem is "polar" to the recently initiated general dual Orlicz-Minkowski problem in [13,15] and "dual" to the newly proposed polar Orlicz-Minkowski problem in [34].

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