The Orlicz version of the L Minkowski problem for −n < p < 0

Orlicz–Minkowski flows

“…(3) The cases (3-b) and (3-d) yield new existence results for the regular even Orlicz-Minkowski problem, while the existence results corresponding to (3-a) and (3-c) can be deduced from [4,32]. (4) Due to the presence of ϕ (possibly non-homogeneous) and ζ in the speed of the flow (2.1), we could not employ the method of [2] to promote subconvergence to full convergence.…”

“…A natural generalization of L p spaces are Orlicz spaces, motivating the Orlicz linear combination of convex bodies and leading to the Orlicz-Minkowski problem. We keep the discussion brief here and the reader may consult [4,[25][26][27]32] as well as [49,54,60,62] for the origin of the Orlicz-Brunn-Minkowski theory and related concepts.…”

On the Musielak-Orlicz-Gauss image problem

“…The Orlicz-Minkowski problem [23] is related to the case when G = t n /n and ϕ is a non-homogeneous function. 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].…”

The $L_p$ Gauss image problem

“…The L 0 Gauss image problem is just the Gauss image problem which was first mentioned in [5], and the existence of smooth solution for the Gauss image problem was in [14]. It is necessary to contrast the L p Gauss image problem with the various Minkowski problems and dual Minkowski problems that have been extensively studied, see [8,11,15,16,27,29,30,34,35,36,37,38,40,44,45] for the L p -Minkowski problem, [7,21,23,24,32,42,43] for the dual Minkowski problem, [6,9,10,25,26,31,39] for the L p dual Minkowski problem, [4,20,22,28] for the Orlicz Minkowski problem, [12,13,18,19,33] for the dual Orlicz Minkowski problem, [17] for the Orlicz Aleskandrov problem.…”