On the general dual Orlicz-Minkowski problem

Existence of smooth even solutions to the dual Orlicz-Minkowski problem

“…The first Orlicz version of the Minkowski problem appeared in [13], at the inception of the Orlicz-Brunn-Minkowski theory in 2010, see [15,17] for recent work on this problem. Later, the dual Minkowski problem was extended to the Orlicz setting and partially solved in [32,33]; here, the q-th dual volume in [16] was replaced by certain dual Orlicz quermassintegrals, while L 0 addition was retained. A common generalization of the problems in [26,32,33] was proposed in [10], in which the q-th dual volume was replaced by a very general notion of dual volume denoted by V ψ (K), and simultaneously L 0 addition was replaced by an extension of L p addition called Orlicz addition.…”

“…Later, the dual Minkowski problem was extended to the Orlicz setting and partially solved in [32,33]; here, the q-th dual volume in [16] was replaced by certain dual Orlicz quermassintegrals, while L 0 addition was retained. A common generalization of the problems in [26,32,33] was proposed in [10], in which the q-th dual volume was replaced by a very general notion of dual volume denoted by V ψ (K), and simultaneously L 0 addition was replaced by an extension of L p addition called Orlicz addition. By combining the general dual volume V ψ (K) with Orlicz addition, a general dual Orlicz curvature measure of K with the origin in its interior denoted by C ϕ, ψ (K, •) is defined in [10], see also [11,12].…”

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

“…If 0 < p < 1, then the Orlicz L p Minkowski problem is due to Jian, Lu [53] if 0 < p < 1. We note that Orlicz versions of the so called L p dual Minkowski are considered recently by Gardner, Hug, Weil, Xing, Ye [28], Gardner, Hug, Xing, Ye [29], Xing, Ye, Zhu [91] and Xing, Ye [92].…”