A Characterization of Convexity-Preserving Maps From a Subset of a Vector Space Into Another Vector Space

Abstract

Let V and X be Hausdorff, locally convex, real, topological vector spaces with dim V > 1. It is shown that a map σ from an open, connected subset of V onto an open subset of X is homeomorphic and convexity‐preserving if and only if σ is projective.

On a characterization of convexity-preserving maps, Davidon’s collinear scalings and Karmarkar’s projective transformations

“…The results in [1,2] are stated and proved in a coordinate-free manner. The first purpose of this note is to state a theorem characterizing continuous, injective, convexitypreserving maps on a relatively open, connected subset of an affine subspace of R m into 1 At first sight the terminology introduced here and in [1, §5] may appear to differ in the following respect: in place of the requirement that σ * is injective on V * stated here, [1, §5] requires that V * = V (i.e.…”

“…7]). However, the characterization proved in [2] is different and apparently new. We refer the reader to [2] for details.…”

“…In [1] the authors have set down a foundation for the subject, stressing those aspects of projective spaces needed in defining and characterizing projective maps on subsets of not-necessarily finite dimensional vector spaces. Based on that foundation, in [2], the authors have shown the equivalence of (homeomorphic) convexity-preserving maps and projective maps on a (connected, open) subset of a (Hausdorff, locally convex, real, topological) vector space (of dimension greater than 1) into another.…”

“…Based on that foundation, in [2], the authors have shown the equivalence of (homeomorphic) convexity-preserving maps and projective maps on a (connected, open) subset of a (Hausdorff, locally convex, real, topological) vector space (of dimension greater than 1) into another. This characterization is related to the classical fundamental theorem of projective geometry [11], and to the more recent characterization of (injective) convexity-preserving maps on an entire real vector space (of dimension greater than 1) into another as affine maps due to Meyer and Kay [10] (see also [12,Chap.…”

“…It also turns out that if we assume that V * = V , then the condition that σ * is injective on V * can be replaced by the condition that σ is injective on D. Thus the requirement that σ * is injective on V * stated here could be replaced without loss of generality by the requirements that {v ∈ V : α(v) = 0, a(v) = 0} is void and that σ is injective on D. The former seems natural since it requires σ * be injective on "where it is defined", while the latter seems less natural because it assumes that {v ∈ V : α(v) = 0, a(v) = 0} is void apriori. Thus we have made the former choice here, while in [1], where connections to projective concepts are clearer, we have opted to use the latter. R n as projective maps (2).…”