Simplicial and Conical Decomposition of Positively Spanning Sets

Abstract

We investigate the decomposition of a set $X$, which positively spans the Euclidean space $\mathbb{R}^{d}$ into a set of minimal positive bases, we call simplices, and into maximal sets positively spanning pointed cones, i.e. cones with exactly one apex. For any set $X$, let $\mathcal{S}(X)$ denote the set of simplex subsets of $X$, and let $\ell(X)$ denote the linear hull of $X$. The set $X$ is said to fulfill the factorisation condition if and only if for each subset $Y\subset X$ and each simplex $S\in\mathcal{S}(X)$, $\ell(Y)\cap\ell(S) = \ell(Y\cap S)$. We demonstrate that $X$ is a positive basis if and only if it is the union of most d simplices, and $X$ satisfies the factorization condition. In this case, $X$ contains a linear basis $B$ such that each simplex in $\mathcal{S}(X)$ has with $B$, all but one exactly one element in common. We show that for sets positively spanning $\mathbb{R}^{d}$, the set of subbases of $X$ forms a boolean lattice, which can be embedded into the set $2^{\mathcal{S}(X)}$, with isomorphy for positive bases. Our second main result depending on the former is as follows. A finite set $X\subset\mathbb{R}^{d}\setminus\{0\}$ can be written as the union of at most $2^{d}$ maximal sets spanning pointed cones, which, if $X$ is a positive basis, are tantamount to frames of the cones. The inequality holds sharply if and only if $X$ is a cross, that is, a union of 1-simplices derived from a linear basis of $\mathbb{R}^{d}$. We also show that there can be at the most $2^{d}$ maximal subsets of $X$ spanning pointed cones, when intersections of two of them do not span a set of full dimension.