Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Lp-Dual geominimal surface area | Weidong and Chen Journal of Inequalities and Applications 2011 2011 6 http content 2011 1 6 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access Lp-Dual geominimal surface area Wang Weidong and Qi Chen Correspondence wdwxh722@ Department of Mathematics China Three Gorges University Yichang 443002 China SpringerOpen0 Abstract Lutwak proposed the notion of Lp-geominimal surface area according to the Lp-mixed volume. In this article associated with the Lp-dual mixed volume we introduce the Lp-dual geominimal surface area and prove some inequalities for this notion. 2000 Mathematics Subject Classification 52A20 52A40. Keywords Lp-geominimal surface area Lp-mixed volume Lp-dual geominimal surface area Lp-dual mixed volume 1 Introduction and main results Let Kn denote the set of convex bodies compact convex subsets with nonempty interiors in Euclidean space R . For the set of convex bodies containing the origin in their interiors and the set of origin-symmetric convex bodies in R we write Kn and Kn respectively. Let son denote the set of star bodies about the origin in R . Let S -1 denote the unit sphere in R denote by V K the -dimensional volume of body K for the standard unit ball B in R denote V B . The notion of geominimal surface area was given by Petty 1 . For K e Kn the geo-minimal surface area G K of K is defined by 1 _ 1 . G K inf nVi K Q V Q n Q e Kn . Here Q denotes the polar of body Q and V1 M N denotes the mixed volume of M N e Kn 2 . According to the Lp-mixed volume Lutwak 3 introduced the notion of Lp-geomini-mal surface area. For K e Kn p 1 the Lp-geominimal surface area Gp K of K is defined by ofpGp K inf nVp K Q V Q n Q e Kn0 . 1 1 Here Vp M N denotes the Lp-mixed volume of M N e 3 4 . Obviously if p 1 Gp K is just the geominimal surface area G K . Further Lutwak 3 proved the following result for the Lp-geominimal surface area. Theorem . Ij K e Kn p 1 the Gp K nJnpV K
