UPPER BOUNDS FOR THE COVERING NUMBER OF CENTRALLY SYMMETRIC CONVEX BODIES IN Rn
نویسندگان
چکیده
The covering number c(K) of a convex body K is the least number of smaller homothetic copies of K needed to cover K . We provide new upper bounds for c(K) when K is centrally symmetric by introducing and studying the generalized α -blocking number βα 2 (K) of K . It is shown that when a centrally symmetric convex body K is sufficiently close to a centrally symmetric convex body K′ , then c(K) is bounded by βα 2 (K ′) from above, where α is a properly chosen number. Related results in Minkowski geometry are also presented. Mathematics subject classification (2010): 52A10, 46B20.
منابع مشابه
On the symmetric average of a convex body
We introduce a new parameter, symmetric average, which measures the asymmetry of a given non-degenerated convex body K in Rn. Namely, sav(K) = infa∈intK ∫ Ka ‖ − x‖Ka dx/|K|, where |K| denotes the volume of K and Ka = K − a. We show that for polytopes sav(K) ≤ C ln N , where N is the number of facets of K. Moreover, in general n n+1 ≤ sav(K) < √ n and equality in the lower bound holds if and on...
متن کاملNakajima’s Problem for General Convex Bodies
For a convex body K ⊂ Rn, the kth projection function of K assigns to any k-dimensional linear subspace of Rn the k-volume of the orthogonal projection of K to that subspace. Let K and K0 be convex bodies in Rn, let K0 be centrally symmetric and satisfy a weak regularity assumption. Let i, j ∈ N be such that 1 ≤ i < j ≤ n − 2 with (i, j) 6= (1, n−2). Assume that K and K0 have proportional ith p...
متن کاملNakajima’s Problem: Convex Bodies of Constant Width and Constant Brightness
For a convex body K ⊂ Rn, the kth projection function of K assigns to any k-dimensional linear subspace of Rn the k-volume of the orthogonal projection of K to that subspace. Let K and K0 be convex bodies in Rn, and let K0 be centrally symmetric and satisfy a weak regularity and curvature condition (which includes all K0 with ∂K0 of class C2 with positive radii of curvature). Assume that K and ...
متن کاملCovering shadows with a smaller volume
For n ≥ 2 a construction is given for convex bodies K and L in R n such that the orthogonal projection Ku can be translated inside Lu for every direction u, while the volumes of K and L satisfy Vn(K) > Vn(L). A more general construction is then given for n-dimensional convex bodies K and L such that the orthogonal projection Kξ can be translated inside Lξ for every k-dimensional subspace ξ of R...
متن کاملCovering Numbers and “low M-estimate” for Quasi-convex Bodies
This article gives estimates on covering numbers and diameters of random proportional sections and projections of symmetric quasi-convex bodies in Rn. These results were known for the convex case and played an essential role in development of the theory. Because duality relations can not be applied in the quasiconvex setting, new ingredients were introduced that give new understanding for the c...
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 2014