tailieunhanh - The origin symmetric for polytopes

The Lp-Blascke addition for a pair of origin symmetric convex bodies is extended to a pair of convex polytopes containing the origin in their interior and Lp Kneser - Suss inequality for polytopes is established. | JOURNAL OF SCIENCE OF HNUE DOI Mathematical and Physical Sci. 2015 Vol. 60 No. 7 pp. 30-34 This paper is available online at http THE ORIGIN SYMMETRIC FOR POLYTOPES Bui Thi Nghia1 Tran Thi Hien2 Dinh Thi Van Khanh2 and Lai Duc Nam2 1 Hoang Quoc Viet Upper Secondary School Yenbai 2 Yen Bai Teacher s Training College Yen Bai Abstract. The Lp -Blascke addition for a pair of origin symmetric convex bodies is extended to a pair of convex polytopes containing the origin in their interior and Lp Kneser - S uss inequality for polytopes is established. Keywords Convex body Blascke addition Lp -Blascke addition polytopes. 1. Introduction The operation between convex bodies now called Blaschke addition goes back to Minkowski 1 117 at least when the bodies are polytopes. Given convex polytopes K and L in Rn a new convex polytope K L called the Blaschke sum of K and L has a facet with a normal outer unit in a given direction if and only if either K or L or both do in which case the area . n 1 -dimensional volume of the facet is the sum of the areas of the corresponding facets of K and L. Blaschke 2 112 found a definition suitable for smooth convex bodies in R3 . The modern definition appropriate for any pair of convex bodies had to wait for the development of surface area measures and is due to Fenchel and Jessen 3 . They defined the surface area measure of K L to be the sum of the surface area measures of K and L and this determines the Blaschke sum up to translation See 1 . The existence of K L is guaranteed by Minkowski s existence theorem a classical result that can be found along with definitions and terminology in Section 2. The Lp -Blaschke addition for any pair of origin symmetric convex bodies was defined by Lutwak 4 by using the solution of the even Lp Minkowski problem. In this paper we extend the Lp -Blaschke addition to any pair of convex polytopes containing the origin in their interior from a pair of origin .