Abstract
In this paper, we study the general dual Orlicz geominimal surface area by the general dual Orlicz mixed volume which was introduced by Gardner et al. (2019). We find the conditions to the existence of the general dual Orlicz-Petty body and hence prove the continuity of the general geominimal surface area in the Orlicz setting (2010 Mathematics Subject Classification: 52A20, 53A15).
Highlights
In the Euclidean space Rn, we call a compact and convex subset K ⊂ Rn a convex body if K has nonempty interior
E geominimal surface area and its extension play important roles in connecting the relative differential geometry with Minkowski geometry. e geominimal surface area was firstly introduced by Petty in 1974. e Lp geominimal surface area was investigated by Lutwak
In 2011, Wang and Qi [12] introduced the Lp geominimal surface area in dual theory which is dual to the Lp geominimal surface area
Summary
In the Euclidean space Rn, we call a compact and convex subset K ⊂ Rn a convex body if K has nonempty interior. E geominimal surface area was firstly introduced by Petty in 1974 (see [1]). The geominimal surface area in Orlicz setting was studied in [8, 9]. E dual Lp Brunn–Minkowski theory was introduced in [10, 11]. E dual Brunn–Minkowski theory in Orlicz setting has already been introduced by Gardner et al [13, 14], Ye [15], and Zhu et al [16]. E Orlicz–Brunn–Minkowski theory for the general volume was established by Gardner et al [21], and the general dual mixed volume V φ,φ(K, L) in Orlicz setting was introduced (see [21], p13), i.e., for two star bodies K and L: V φ,φ(K, L).
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