Finite-dimensional lattice-subspaces of $C(\Omega )$ and curves of $\mathbb {R}^n$

Abstract:

Let $x_1,\dotsc ,x_n$ be linearly independent positive functions in $C(\Omega )$, let $X$ be the vector subspace generated by the $x_i$ and let $\beta$ denote the curve of $\mathbb R^n$ determined by the function $\beta (t)=\frac {1}{z(t)} (x_1(t),x_2(t),\dotsc ,x_n(t))$, where $z(t)=x_1(t)+x_2(t)+\dotsb +x_n(t)$. We establish that $X$ is a vector lattice under the induced ordering from $C(\Omega )$ if and only if there exists a convex polygon of $\mathbb R^n$ with $n$ vertices containing the curve $\beta$ and having its vertices in the closure of the range of $\beta$. We also present an algorithm which determines whether or not $X$ is a vector lattice and in case $X$ is a vector lattice it constructs a positive basis of $X$. The results are also shown to be valid for general normed vector lattices.