Prompted by the development of algorithms for analysing geometric tolerancing, this article describes a method to determine the Minkowski sum for 3-dimension polytopes. This purposed method is based exclusively on intersection operations on normal cones, using the properties of the normal fan of a Minkowski sum obtained by common refinement of the normal fans of the operands. It can be used to ...

We introduce and study the family of sets in a finite dimensional Euclidean space which can be written as the Minkowski sum of a compact and convex set and a convex cone (not necessarily closed). We establish several properties of the class of such sets, called Motzkin predecomposable, some of which hold also for the class of Motzkin decomposable sets (i.e., those for which the convex cone in t...

In this paper we review the existing linear and quadratic complexity (upper) bounds on the values of the positive roots of polynomials and their impact on the performance of the Vincent-Akritas-Strzeboński (VAS) continued fractions method for the isolation of real roots of polynomials. We first present the following four linear complexity bounds (two “old” and two “new” ones, respectively): Cau...

A formal de nition of morphological operators in (max,min)algebra is introduced and their relevant properties from an algebraic viewpoint are stated. Some previous works in mathematical morphology have already encountered this type of operators but a systematic study of them has not yet been undertaken in the morphological literature. It is shown in particular that one of their fundamental prop...

Let A + B be the pointwise (Minkowski) sum of two convex subsets A and B of a Banach space. Is it true that every continuous mapping h : X → A + B splits into a sum h = f + g of continuous mappings f : X → A and g : X → B? We study this question within a wider framework of splitting techniques of continuous selections. Existence of splittings is guaranteed by hereditary invertibility of linear ...

Let P be a simple polygon with m edges, which is the disjoint union of k simple polygons, all monotone in a common direction u, and let Q be another simple polygon with n edges, which is the disjoint union of l simple polygons, all monotone in a common direction v. We show that the combinatorial complexity of the Minkowski sum P ⊕ Q is O(klmnα(min{m,n})). Some structural properties of the case ...

We extend the class of reversal-bounded counter machines by authorizing a finite number of alternations between increasing and decreasing mode over a given bound. We prove that extended reversal-bounded counter machines also have effective semi-linear reachability sets. We also prove that the property of being reversal-bounded is undecidable in general even when we fix the bound, whereas this p...