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For origin-symmetric convex bodies (i.e., the unit balls of finite dimensional Banach spaces) it is conjectured that there exist a family of inequalities each of which is stronger than the classical Brunn-Minkowski inequality and a family of inequalities each of which is stronger than the classical Minkowski mixed-volume inequality. It is shown that these… (More)

- Károly J. Böröczky, Imre Z. Ruzsa
- Discrete & Computational Geometry
- 2007

G. Wegner [12] gave a geometric characterization of all so–called Groemer packing of n ≥ 2 unit discs in E2 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 proportion… (More)

Let C be a convex body in Euclidean d-space IE , i.e., a compact convex set with non-empty interior, and denote by P i n and P (n) the set of polytopes with at most n vertices inscribed to C and the set of polytopes with at most n facets circumscribed to C, respectively. Denote by δ(., .) the symmetric difference metric. Beginning with the work of L. Fejes… (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)

In analogy with the classical Minkowski problem, necessary and sufficient conditions are given to assure that a given measure on the unit sphere is the cone-volume measure of the unit ball of a finite dimensional Banach space.

- Károly J. Böröczky, Ferenc Fodor, Matthias Reitzner, V. Vígh
- J. Multivariate Analysis
- 2009

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)

How well a polytope of restricted complexity can approximate a smooth convex body in R? This natural question has attracted the attention of mathematicians of various background since the middle of the 20th century. In this extended abstract, polytopes are always inscribed, and restricted complexity mostly means restricting the number of vertices of the… (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)

We verify a conjecture of Lutwak, Yang, Zhang about the equality case in the Orlicz-Petty projection inequality, and provide an essentially optimal stability version. The Petty projection inequality (Theorem 1), its Lp extension, and its analytic counterparts, the Zhang-Sobolev inequality [43] and its Lp extension by A. Cianchi, E. Lutwak, D. Yang, G. Zhang… (More)