Vector-Lattices and a Problem in Geometry

Abstract

Let X be a subset of ℝⁿ and suppose f:X → ℝ is “weakly piecewise linear” in the following sense: there is a finite collection of closed piecewise linear sets {P_i} such that X ⊆ UP_i and for each i, the restriction of f to X ∩ P_i is (affine) linear. Is f “strictly piecewise linear”, i.e., is f the restriction to X of a piecewise linear function defined on ℝⁿ ? In general the answer is no, though it is not hard to think of conditions on X which are sufficient for a yes answer, e.g., X itself is a closed p.1. set. We show in this paper that there is a geometric condition on the embedding X ⊆ ℝⁿ which is necessary and sufficient for every weakly p.1. f:X → ℝ to be strictly p.1.. It is a kind of local connectedness near the points of the remainder of a certain non-Hausdorff completion of X. The proof involves facts about vector-lattice ideals and, implicitly, the duality between finitely presented vector-lattices and piecewise linear sets and maps due to Beynon (1975) and anticipated by Baker (1968). While the Baker-Beynon theory has been used in the past to derive results about vector-lattices from well known geometric facts, the present paper is the first instance known to the author in which it is used to derive a theorem in elementary geometry from algebraic facts about vector-lattices.