L_p-Polar Projection Brunn-Minkowski Inequality

Abstract

In this paper,the authors first generalize the notion of classical dual mixed volume to L_p-space and introduce the notion of q-dual mixed volume.Moreover,they extend the notion of classical L_p(p≥1)-projection bodies and introduce the notions of L_p(p<l)-projection and mixed projection bodies,and establish the Brunn-Minkowski inequality for L_p-polar mixed projection bodies.As applications,the well-known Brunn-Minkowski inequality for polar of projection bodies is generalized and an L_p-polar form of Brunn-Minkowski inequality for projection bodies is obtained.

將經典的對偶混合體積概念推廣到L_p空間,提出了'q-全對偶混合體積'的概念.將傳統的p≥1的L_p投影體概念拓展,提出p<1時的L_p投影體和混合投影體概念,并且建立了L_p-極投影Brunn-Minkowski不等式.作為應用,推廣了熟知的極投影Brunn-Minkowski不等式,獲得了投影Brunn-Minkowski不等式的L_p空間的極形式.