A close discrete analog of the classical Brunn-Minkowksi inequality that holds for finite subsets of the integer lattice is obtained. This is applied to obtain strong new lower bounds for the cardinality of the sum of two finite sets, one of which has full dimension, and, in fact, a method for computing the exact lower bound in this situation, given the dimension of the lattice and the cardinal...

In noncommutative probability theory independence can be based on free products instead of tensor products. This yields a highly noncommutative theory: free probability (for an introduction see [9]). The analogue of entropy in the free context was introduced by the second named author in [8]. Here we show that Shannon's entropy power inequality ([6],[1]) has an analogue for the free entropy χ(X...

There exists an extensive literature on Wiener sausages; see e.g. [25] and the references therein. Nevertheless, relatively little is known so far about the geometry of Wiener sausages due to the complex nature of their realizations. One possible description of the geometric structure of (sufficiently regular) subsets of R is given by their d + 1 intrinsic volumes or Minkowski functionals inclu...

The combinatorial complexities of (1) the Voronoi diagram of moving points in 2D and (2) the Voronoi diagram of lines in 3D, both under the Euclidean metric, continues to challenge geometers because of the open gap between the (n2) lower bound and the O(n3+ε) upper bound. Each of these two combinatorial problems has a closely related problem involving Minkowski sums: (1′) the complexity of a Mi...

Let P,Q ⊆ R be two n-point multisets and Ar ≥ b be a set of λ inequalities on x and y, where A ∈ R, r = [y ], and b ∈ R. Define the constrained Minkowski sum (P ⊕Q)Ar≥b as the multiset {(p+ q)|p ∈ P, q ∈ Q,A(p+ q) ≥ b}. Given P , Q, Ar ≥ b, an objective function f : R → R, and a positive integer k, the Minkowski Sum Selection problem is to find the k largest objective value among all objective ...

Geometrical patterns are encountered in many different fields of science and technology, including porous media, biology, polymer chemistry, metallography, geology and astronomy. 1), 2) We characterize these patterns in terms of shape (geometry) and connectivity (topology) by means of morphological image analysis (MIA). This involves the calculation of the Minkowski functionals known from integ...

In our contribution we study cubic and quintic Pythagorean Hodograph (PH) curves in the Euclidean and Minkowski planes. We analyze their control polygons and give necessary and sufficient conditions for cubic and quintic curves to be PH. In the case of Euclidean cubics the conditions are known and we provide a new proof. For the case of Minkowski cubics we formulate and prove a new simple geome...

We introduce a notion of the Euclidean and the Minkowski rank for arbitrary metric spaces and we study their behaviour with respect to products. We show that the Minkowski rank is additive with respect to metric products, while additivity of the Euclidean rank does not hold in general.

In this paper we study properties of sections of convex bodies with respect to the Gaussian measure. We develop a formula connecting the Minkowski functional of a convex symmetric body K with the Gaussian measure of its sections. Using this formula we solve an analog of the Busemann-Petty problem for Gaussian measures.

In this paper we introduce and investigate similarity measures for convex polyhedra based on Minkowski addition and inequalities for the mixed volume, volume and surface area related to the Brunn-Minkowski theory. All measures considered are invariant under translations; furthermore, they may also be invariant under subgroups of the aane transformation group. For the case of rotation and scale ...