Isomorphic Properties of Intersection Bodies
نویسندگان
چکیده
We study isomorphic properties of two generalizations of intersection bodies the class I k of k-intersection bodies in R and the class BPk of generalized k-intersection bodies in R. In particular, we show that all convex bodies can be in a certain sense approximated by intersection bodies, namely, if K is any symmetric convex body in R and 1 6 k 6 n− 1 then the outer volume ratio distance from K to the class BPk can be estimated by o.v.r.(K,BPk ) := inf{ ( |C| |K| ) 1 n : C ∈ BPk , K ⊆ C} 6 c √ n k log en k , where c > 0 is an absolute constant. Next we prove that if K is a symmetric convex body in R, 1 6 k 6 n− 1 and its k-intersection body Ik(K) exists and is convex, then dBM (Ik(K), B n 2 ) 6 c(k), where c(k) is a constant depending only on k, dBM is the Banach-Mazur distance, and B 2 is the unit Euclidean ball in R. This generalizes a well-known result of Hensley and Borell. We conclude the paper with volumetric estimates for k-intersection bodies.
منابع مشابه
Volume difference inequalities for the projection and intersection bodies
In this paper, we introduce a new concept of volumes difference function of the projection and intersection bodies. Following this, we establish the Minkowski and Brunn-Minkowski inequalities for volumes difference function of the projection and intersection bodies.
متن کاملComplex intersection bodies
We introduce complex intersection bodies and show that their properties and applications are similar to those of their real counterparts. In particular, we generalize Busemann’s theorem to the complex case by proving that complex intersection bodies of symmetric complex convex bodies are also convex. Other results include stability in the complex Busemann-Petty problem for arbitrary measures an...
متن کاملHamilton Decompositions of Block-Intersection Graphs of Steiner Triple Systems
Block-intersection graphs of Steiner triple systems are considered. We prove that the block-intersection graphs of non-isomorphic Steiner triple systems are themselves non-isomorphic. We also prove that each Steiner triple system of order at most 15 has a Hamilton decomposable block-intersection graph.
متن کاملSOME RESULTS ON THE COMPLEMENT OF THE INTERSECTION GRAPH OF SUBGROUPS OF A FINITE GROUP
Let G be a group. Recall that the intersection graph of subgroups of G is an undirected graph whose vertex set is the set of all nontrivial subgroups of G and distinct vertices H,K are joined by an edge in this graph if and only if the intersection of H and K is nontrivial. The aim of this article is to investigate the interplay between the group-theoretic properties of a finite group G and the...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2011