Random approximation and the vertex index of convex bodies
نویسندگان
چکیده
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...
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Assume K ⊂ R is a convex body and Xn ⊂ K is a random sample of n uniform, independent points from K. The convex hull of Xn is a convex polytope Kn called random polytope inscribed in K. We are going to investigate various properties of this polytope: for instance how well it approximates K, or how many vertices and facets it has. It turns out that Kn is very close to the so called floating body...
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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...
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Choose n independent random points on the boundary of a convex body K ⊂Rd . The intersection of the supporting halfspaces at these random points is a random convex polyhedron. The expectations of its volume, its surface area and its mean width are investigated. In the case that the boundary of K is sufficiently smooth, asymptotic expansions as n→∞ are derived even in the case when the curvature...
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ژورنال
عنوان ژورنال: Archiv der Mathematik
سال: 2016
ISSN: 0003-889X,1420-8938
DOI: 10.1007/s00013-016-0975-2