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- Author or Editor: Károly Böröczky x
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Abstract
A convex d-polytope in ℝ d is called edge-antipodal if any two vertices that determine an edge of the polytope lie on distinct parallel supporting hyperplanes of the polytope. We introduce a program for investigating such polytopes, and examine those that are simple.
A version of the celebrated Moment Theorem of László Fejes Tóth is proved where the integrand is based not on the second moment but on another quadratic form.
Let a k denote the maximum number with the property that one can place k points on the unit sphere S 2 so that the spherical distance between any two different points is at least a k. The exact value of a k is determined only for some small values of k, namely, for k = 12 and k=24. In this paper we give new upper bounds on a k for k=14,15,16,17.
Let C be a convex body in the Euclidean plane. By the relative distance of points p and q we mean the ratio of the Euclidean distance of p and q to the half of the Euclidean length of a longest chord of C parallel to pq. In this note we find the least upper bound of the minimum pairwise relative distance of six points in a plane convex body.
A star body (with respect to the origin 0) in ℝ d ( d ≧ 3) which has 0 as center of symmetry is uniquely determined by the ( d − 1)-dimensional volumes of its sections with hyperplanes through 0. Without the symmetry assumption, we show that a star body is uniquely determined by the volumes and centroids of its hyperplane sections through 0. For convex bodies, we prove a stability version of this result.
Summary
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Summary
The notion of successive illumination parameters of convex bodies is introduced. We prove some theorems in the plane and determine the exact values of the successive illumination parameters of spheres, cubes and cross-polytopes for some dimensions.
Abstract
Let K be a convex body in ℝ d , let j ∈ {1, …, d−1}, and let K(n) be the convex hull of n points chosen randomly, independently and uniformly from K. If ∂K is C + 2, then an asymptotic formula is known due to M. Reitzner (and due to I. Bárány if ∂K is C + 3) for the difference of the jth intrinsic volume of K and the expectation of the jth intrinsic volume of K(n). We extend this formula to the case when the only condition on K is that a ball rolls freely inside K.
Extending Blaschke and Lebesgue’s classical result in the Euclidean plane, it has been recently proved in spherical and the hyperbolic cases, as well, that Reuleaux triangles have the minimal area among convex domains of constant width D. We prove an essentially optimal stability version of this statement in each of the three types of surfaces of constant curvature. In addition, we summarize the fundamental properties of convex bodies of constant width in spaces of constant curvature, and provide a characterization in the hyperbolic case in terms of horospheres.
Summary
We determine the minimal radius of