G. Wegner  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)
We characterize the duality of convex bodies in d-dimensional Euclidean vector space, viewed as a mapping from the space of convex bodies containing the origin in the interior into the same space. The question for such a characterization was posed by Vitali Milman. Sufficient for a characterization, up to a trivial exception and the composition with a… (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)
The optimal condition of the cone volume measure of a pair of antipodal points is proved and analyzed.