Károly J. Böröczky

The Mean Width of Circumscribed Random Polytopes

For a given convex body K in Rd, 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 simplicial polytope P, a precise asymptotic formula for the difference of the mean widths of P(n) and P is obtained.

Maximal volume enclosed by plates and proof of the chessboard conjecture

and FT, resp., and ~{Vol,_,~~~F;u}~~{Vol,_,F~/F~~v} holds for every UE[W”, then Vol P’IVol P < 2”‘(“-‘). So me related questions are also considered.

Expectation of intrinsic volumes of random polytopes

Let K be a convex body in ℝd, 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.

The mean width of random polytopes circumscribed around a convex body

Let K be a d-dimensional convex body and let K(n) be the intersection of n halfspaces containingnK whose bounding hyperplanes are independent and identically distributed. Under suitablendistributional assumptions, we prove an asymptotic formula for the expectation of the differencenof the mean widths of K(n) and K, and another asymptotic formula for the expectation of thennumber of facets of K(n). These results are achieved by establishing an asymptotic result onnweighted volume approximation of K and by �dualizing� it using polarity.

On the successive illumination parameters of convex bodies

SummaryThe notion of successive illumination parameters of convex bodies isnintroduced. We prove some theorems in the plane and determine the exact valuesnof the successive illumination parameters of spheres, cubes and cross-polytopesnfor some dimensions.

About the error term for best approximation with respect to the Hausdorff related metrics

Let M be a convex body with C+3 boundary in ℝd, d ≥ 3, and consider a polytope Pn (or P(n)) with at most n vertices (at most n facets) minimizing the Hausdorff distance from M. It has long been known that as n tends to infinity, there exist asymptotic formulae of order n−2/(d-1) for the Hausdorff distances δH(Pn, M) and δH(P(n), M). In this paper a bound of order n−5/(2(d-1)) is given for the error of the asymptotic formulae.This bound is clearly not the best possible, and Gruber[9] conjectured that if the boundary of M is sufficiently smooth, then there exist asymptotic expansions for δH(Pn, M) and δH(P(n), M). With the help of quasiconformal mappings, we show for the three-dimensional unit ball that the error is at least f (n) · n−2 where f (n) tends to infinity. Therefore in this case, no asymptotic expansion exists in terms of n−2/(d-1) = n−1.

On the cardinality of sumsets in torsion-free groups

Let

Polytopes of Minimal Volume with Respect to a Shell—Another Characterization of the Octahedron and the Icosahedron

A, B

Valuations on Lattice Polytopes

be finite subsets of a torsion-free group

Note on an Inequality of Wegner

G