Abstract
A new concept of p- Aleksandrov body is firstly introduced. In this paper, p- Brunn-Minkowski inequality and p- Minkowski inequality on the p- Aleksandrov body are established. Furthermore, some pertinent results concerning the Aleksandrov body and the p- Aleksandrov body are presented. 2000 Mathematics Subject Classification: 52A20 52A40
Highlights
The notion of Aleksandrov body was firstly introduced by Aleksandrov to solve Minkowski problem in 1930s in [1]
The Aleksandrov body establishes the relationship between the convex body containing the origin and the positive continuous functions and characterizes the convex body by means of the positive continuous functions
P-Brunn-Minkowski inequality and p-Minkowski inequality for the p-Aleksandrov bodies associated with positive continuous functions are established
Summary
The notion of Aleksandrov body was firstly introduced by Aleksandrov to solve Minkowski problem in 1930s in [1]. P-Brunn-Minkowski inequality and p-Minkowski inequality for the p-Aleksandrov bodies associated with positive continuous functions are established. Some related results, including of the uniqueness results, the convergence results for the Aleksandrov bodies and the p-Aleksandrov bodies associated with positive continuous functions, are presented. Let V (K) denote the n-dimensional volume of body K, for the standard unit ball B in Rn, denote ωn = V (B), and let Sn-1 denote the unit sphere in Rn. Let C+(Sn-1) denote the set of positive continuous functions on Sn-1, endowed with the topology derived from the max norm. Following Aleksandrov (see [18]), define the volume V (f) of a function f as the volume of the Aleksandrov body associated with the positive continuous function f. The other aim of this paper is to establish the following inequality for the Aleksandrov bodies and the p-Aleksandrov bodies associated with positive continuous functions. Definitions, and their background materials are exhibited
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