منابع مشابه
On the vertex index of convex bodies
We introduce the vertex index, vein(K), of a given centrally symmetric convex body K ⊂ Rd, which, in a sense, measures how well K can be inscribed into a convex polytope with small number of vertices. This index is closely connected to the illumination parameter of a body, introduced earlier by the first named author, and, thus, related to the famous conjecture in Convex Geometry about covering...
متن کاملA remark on vertex index of the convex bodies
The vertex index of a symmetric convex body K ⊂ Rn, vein(K), was introduced in [BL]. Bounds on the vertex index were given in the general case as well as for some basic examples. In this note we improve these bounds and discuss their sharpness. We show that vein(K) ≤ 24n, which is asymptotically sharp. We also show that the estimate n3/2 √ 2πe ovr(K) ≤ vein(K), obtained in [BL] (here ovr(K) den...
متن کاملOn the covering index of convex bodies
Covering a convex body by its homothets is a classical notion in discrete geometry that has resulted in a number of interesting and long-standing problems. Swanepoel introduced the covering parameter of a convex body as a means of quantifying its covering properties. In this paper, we introduce two relatives of the covering parameter called covering index and weak covering index, which upper bo...
متن کاملAsymmetry of Convex Polytopes and Vertex Index of Symmetric Convex Bodies
In (Gluskin, Litvak in Geom. Dedicate 90:45–48, 2002) it was shown that a polytope with few vertices is far from being symmetric in the Banach–Mazur distance. More precisely, it was shown that Banach–Mazur distance between such a polytope and any symmetric convex body is large. In this note we introduce a new, averaging-type parameter to measure the asymmetry of polytopes. It turns out that, su...
متن کاملThe Covering Index of Convex Bodies
Covering a convex body by its homothets is a classical notion in discrete geometry that has resulted in a number of interesting and long-standing problems. Swanepoel [Mathematika 52 (2005), 47–52] introduced the covering parameter of a convex body as a means of quantifying its covering properties. In this paper, we introduce a relative of the covering parameter called covering index, which turn...
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ژورنال
عنوان ژورنال: Advances in Mathematics
سال: 2007
ISSN: 0001-8708
DOI: 10.1016/j.aim.2007.04.016