Dual Orlicz geominimal surface area

Abstract

The L p -geominimal surface area was introduced by Lutwak in 1996, which extended the important concept of the geominimal surface area. Recently, Wang and Qi defined the p-dual geominimal surface area, which belongs to the dual Brunn-Minkowski theory. In this paper, based on the concept of the dual Orlicz mixed volume, we extend the dual geominimal surface area to the Orlicz version and give its properties. In addition, the isoperimetric inequality, a Blaschke-Santaló type inequality, and the monotonicity inequality for the dual Orlicz geominimal surface areas are established. MSC: 52A30; 52A40

The General Dual Orlicz Geominimal Surface Area

“…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]. For more results on the geominimal surface area, one can refer to [3,4,12,[17][18][19][20] and so on.…”

Dual Orlicz mixed geominimal surface areas

“…Remark 1.2. If i = 0 in Definition 1.1, then (3) gives the dual Orlicz geominimal surface area G φ (K) of K ∈ S n o which was defined by Ma and Wang (see [23]), but it should be noted that φ in (3) is different from φ in [23].…”

DualLp-Mixed Geominimal Surface Area and Related Inequalities

“…Recently, Wang and Qi [5] introduced a concept of dual -geominimal surface area, which is a dual concept for -geominimal surface area and belongs to the dual -Brunn-Minkowski theory for star bodies also developed by Lutwak (see [6,7]). The dual -Brunn-Minkowski theory for star bodies and a more extensive dual Orlicz-Brunn-Minkowski theory for star bodies received considerable attention (see, e.g., [8][9][10][11][12][13][14][15][16][17][18][19][20][21]), and they have been proved to be very powerful in solving many geometric problems, for instance, the Busemann-Petty problems (see, e.g., [6,[22][23][24]).…”