Minkowski sums of simplices in Rn form an interesting class of polytopes that seem to emerge in various situations. In this paper we discuss the Minkowski sum of the simplices ∆k−1 in Rn where k and n are fixed, their flags and some of their face lattice structure. In particular, we derive a closed formula for their exponential generating flag function. These polytopes are simple, include both ...

In this paper, we introduce the notion of a constrained Minkowski sum which for two (finite) point-sets P,Q ⊆ R2 and a set of k inequalities Ax > b is defined as the point-set (P⊕Q)Ax>b = {x = p+q | p ∈ P, q ∈ Q, Ax > b}. We show that typical interval problems from computational biology can be solved by computing a set containing the vertices of the convex hull of an appropriately constrained M...

Let A and B be two convex polytopes in R 3 with m and n facets, respectively. The penetration depth of A and B, denoted as (A; B), is the minimum distance by which A has to be translated so that A and B do not intersect. We present a randomized algorithm that computes (A; B) in O(m 3=4+" n 3=4+" + m 1+" + n 1+") expected time, for any constant " > 0. It also computes a vector t such that ktk = ...

Following a recently considered generalization of linear equations to unordered data vectors, we perform a further generalization to ordered data vectors. These generalized equations naturally appear in the analysis of vector addition systems (or Petri nets) extended with ordered data. We show that nonnegative-integer solvability of linear equations is computationally equivalent (up to an expon...

The similarity of two convex polyhedra A and B may be calculated by evaluating the volume or mixed volume of their Minkowski sum over a specific set of relative orientations. The relative orientations are characterized by the fact that faces and edges of A and B are parallel as much as possible. For one of these relative orientations the similarity measure is optimal. In this article we propose...

In this paper the concept of set invariance for time-delay systems is introduced with a specific attention to the linear discrete-time case. We are interested in the definition of a D(elay)invariant set with respect to a bounded polyhedral subset of the state-space. D-invariance conditions are derived based on the Minkowski addition in a first stage, and subsequently translated in feasibility-b...

The metamorphosis between two user-specified objects offers an intuitive metaphor for designing animations of deforming shapes. We present a new technique for interactively editing such deformations and for animating them in realtime. Besides the starting and ending shapes, our approach offers easy to use additional control over the deformations. The new Bezier Interpolating Polyhedron (BIP) pr...