Károly J. Böröczky

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G. Wegner [12] gave a geometric characterization of all so–called Groe-mer packing of n ≥ 2 unit discs in E 2 that are densest packings of n unit discs with respect to the convex hull of the discs. In this paper we provide a number theoretic characterization of all n satisfying that such a " Wegner packing " of n unit discs exists, and show that the(More)
A stability version of the Blaschke-Santaló inequality and the affine isoperimetric inequality for convex bodies of dimension n ≥ 3 is proved. The first step is the reduction to the case when the convex body is o-symmetric and has axial rotational symmetry. This step works for related inequalities compatible with Steiner symmetrization. Secondly, for these(More)
Two consequences of the stability version of the one dimensional Prékopa-Leindler inequality are presented. One is the stability version of the Blaschke-Santaló inequality, and the other is a stability version of the Prékopa-Leindler inequality for even functions in higher dimensions, where a recent stability version of the Brunn-Minkowski inequality is(More)
Let K be a d-dimensional convex body, and let K (n) be the intersection of n halfspaces containing K whose bounding hyperplanes are independent and identically distributed. Under suitable distributional assumptions, we prove an asymptotic formula for the expectation of the difference of the mean widths of K (n) and K, and another asymptotic formula for the(More)
For a given convex body K in R d , a random polytope K (n) is defined (essentially) as the intersection of n independent closed halfspaces containing K and having an isotropic and (in a specified sense) uniform distribution. We prove upper and lower bounds of optimal orders for the difference of the mean widths of K (n) and K as n tends to infinity. For a(More)