The objective of this paper is to present two types of results on Minkowski sums of convex polytopes. The first is about a special class of polytopes we call perfectly centered and the combinatorial properties of the Minkowski sum with their own dual. In particular, we have a characterization of face lattice of the sum in terms of the face lattice of a given perfectly centered polytope. Exact f...

Given a polygonal shape Q with n vertices, can it be expressed, up to a tolerance ε in Hausdorff distance, as the Minkowski sum of another polygonal shape with a disk of fixed radius? If it does, we also seek a preferably simple solution shape P; P’s offset constitutes an accurate, vertex-reduced, and smoothened approximation of Q. We give a decision algorithm for fixed radius in O(n logn) time...

Probabilistic and deterministic planners are two major approximate-based frameworks for solving motion planning problems. Both approaches have their own advantages and disadvantages. In this work, we provide an investigation to the following question: Is there a planner that can take the advantages from both probabilistic and deterministic planners? Our strategy to achieve this goal is to use t...

AIMS To determine the influence of single-vision lenses (SVLs) and progressive addition lenses (PALs) on the near vision posture of myopic children based on their near phoria. METHODS Sixty-two myopic children were assigned to wear SVLs followed by PALs. Eighteen children were esophoric (greater than +1), 18 were orthophoric (-1 to 1) and 26 were exophoric (less than -1) at near. Reading dist...

Let X1, . . . , XN denote N independent d-dimensional Lévy processes, and consider the N -parameter random field X(t) := X1(t1) + · · ·+XN (tN ). First we demonstrate that for all nonrandom Borel sets F ⊆ R, the Minkowski sum X(R+ ) ⊕ F , of the range X(R+ ) of X with F , can have positive d-dimensional Lebesgue measure if and only if a certain capacity of F is positive. This improves our earli...

We derive tight expressions for the maximum number of k-faces, 0 ≤ k ≤ d − 1, of the Minkowski sum, P1+⋯+Pr, of r convex d-polytopes P1, . . . , Pr in R, where d ≥ 2 and r < d, as a (recursively defined) function on the number of vertices of the polytopes. Our results coincide with those recently proved by Adiprasito and Sanyal [1]. In contrast to Adiprasito and Sanyal’s approach, which uses to...

We express the matroid polytope PM of a matroid M as a signed Minkowski sum of simplices, and obtain a formula for the volume of PM . This gives a combinatorial expression for the degree of an arbitrary torus orbit closure in the Grassmannian Grk,n. We then derive analogous results for the independent set polytope and the associated flag matroid polytope of M . Our proofs are based on a natural...