In this paper, we address the problem of deciding absolute irreducibility of multivariate polynomials. Our work has been motivated by a recent work due to Gao et. al. [1, 2, 3] where they have considered the problem for bivariate polynomials by studying the integral decomposability of polygons in the sense of Minkowski sum. We have generalized their result to polynomials containing arbitrary nu...

Hedgehogs are (possibly singular and self-intersecting) hypersurfaces that describe Minkowski differences of convex bodies in R. They are the natural geometrical objects when one seeks to extend parts of the Brunn-Minkowski theory to a vector space which contains convex bodies. In terms of characteristic functions, Minkowski addition of convex bodies correspond to convolution with respect to th...

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 ...

We find conditions ensuring the existence of the one-sided Minkowski content for d-dimensional closed sets in R, in connection with regularity properties of their boundaries. Moreover, we provide a class of sets stable under finite unions for which the one-sided Minkowski content exists. It follows, in particular, that finite unions of sets with Lipschitz boundary and a type of sets with positi...

We present an algorithm for computing Minkowski sums among surfaces of revolution and surfaces of linear extrusion, generated by slope-monotone closed curves. The special structure of these simple surfaces allows the process of normal matching between two surfaces to be expressed as an explicit equation. Based on this insight, we also present an efficient algorithm for computing the distance be...

The objective of this paper is to develop a set of reliable methods to build confidence sets for the Aumann mean of a random closed set estimated through the Minkowski empirical mean. In order to do so, we introduce a procedure to build a confidence set based on Weil’s result for the Hausdorff distance between the empirical and the Aumann means; then, we introduce another procedure based on the...

We show that for polytopes P1, P2, . . . , Pr ⊂ R, each having ni ≥ d + 1 vertices, the Minkowski sum P1 + P2 + · · · + Pr cannot achieve the maximum of ∏ i ni vertices if r ≥ d. This complements a recent result of Fukuda & Weibel (2006), who show that this is possible for up to d − 1 summands. The result is obtained by combining methods from discrete geometry (Gale transforms) and topological ...

It is known that in the Minkowski sum of r polytopes in dimension d, with r < d, the number of vertices of the sum can potentially be as high as the product of the number of vertices in each summand [2]. However, the number of vertices for sums of more polytopes was unknown so far. In this paper, we study sums of polytopes in general orientations, and show a linear relation between the number o...

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We consider the problem of traveling the contour of the set of all points that are within distance 1 of a connected planar curve arrangement P, forming an embedding of the graph G. We show that if the overall length of P is L, there is a closed roundtrip that visits all points of the contour and has length no longer than 2L + 2n. This result carries over in a more general setting: if R is a com...