Barycentric coordinates are unique for triangles, but there are many possible generalizations to convex polygons. In this paper we derive sharp upper and lower bounds on all barycentric coordinates over convex polygons and use them to show that all such coordinates have the same continuous extension to the boundary. We then present a general approach for constructing such coordinates and use it...

Many recent works attempt to generalize barycentric coordinates to arbitrary polygons. I construct a counterexample proving that no such generalization will produce purely bijective mappings in the plane provided the coordinates meet the Lagrange, reproduction, and partition of unity properties. The proof concerns generalized barycentric coordinates in a square, but trivially generalizes to arb...

We derive some new results on the k-th barycentric Olson constants of abelian groups (mainly cyclic). This quantity, for a finite abelian (additive) group (G,+), is defined as the smallest integer l such that each subset A of G with at least l elements contains a subset with k elements {g1, . . . , gk} satisfying g1 + · · ·+ gk = k gj for some 1 ≤ j ≤ k.

For a simplicial complex or more generally Boolean cell complex ∆ we study the behavior of the f and h-vector under barycentric subdivision. We show that if ∆ has a non-negative h-vector then the h-polynomial of its barycentric subdivision has only simple and real zeros. As a consequence this implies a strong version of the Charney-Davis conjecture for spheres that are the subdivision of a Bool...

Let ∆ be a Euclidean n-simplex and let {∆j} denote a finite union of simplices which partition ∆. We assume that the partition is invariant under the affine symmetry group of ∆. A classical example of such a partition is the one obtained from barycentric subdivision, but there are plenty of other possibilities. (See §4.1, or else [Sp, p. 123], for a definition of barycentric subdivision.) Our p...

In this article we describe the notion of affinely independent subset of a real linear space. First we prove selected theorems concerning operations on linear combinations. Then we introduce affine independence and prove the equivalence of various definitions of this notion. We also introduce the notion of the affine hull, i.e. a subset generated by a set of vectors which is an intersection of ...

For simplicity, we adopt the following rules: x denotes a set, n, m, k denote natural numbers, r denotes a real number, V denotes a real linear space, v, w denote vectors of V , A1 denotes a finite subset of V , and A2 denotes a finite affinely independent subset of V . One can prove the following propositions: (1) For all real-valued finite sequences f1, f2 and for every real number r holds (I...

With the methods of barycentric coordinates, we establish several collinearities in the heptagonal triangle, formed by a side and two diagonals of different lengths of a regular heptagon. 1. The regular heptagon Consider a regular heptagon AA′C ′A′′BCB′ inscribed in a circle, each side of length a. The diagonals are of two kinds. Those with 3 vertices on the defining minor arc have the same len...

Contemporary pulsar timing experiments have reached a sensitivity level where systematic errors introduced by existing analysis procedures are limiting the achievable science. We have developed tempo2, a new pulsar timing package that contains propagation and other relevant effects implemented at the 1 ns level of precision (a factor of ∼ 100 more precise than previously obtainable). In contras...