On a characterization of convexity-preserving maps, Davidon's collinear scalings and Karmarkar's projective transformations (Q5930729)

In a recent paper, the authors have proved results characterizing convexity-preserving maps defined on a subset of a not necessarily finite-dimensional real vector space as projective maps. In this paper, the authors first state a theorem characterizing continuous, injective, convexity-preserving maps from a relatively open, connected subset of an affine subspace of \(\mathbb{R}^m\) into \(\mathbb{R}^n\) as projective maps. Second, based on that characterization theorem, the authors offer a characterization theorem for collinear scalings first introduced by \textit{W. C. Davidon} [SIAM J. Numer. Anal. 17, 268-281 (1980; Zbl 0242.65026)] for deriving certain algorithms for nonlinear optimization, and a characterization theorem for projective transformations used by \textit{N. Karmarkar} [Combinatorica 4, 373-395 (1984; Zbl 0557.90065)] in his linear programming algorithm. The proofs of these two theorems utilize the authors' characterization of continuous, injective, convexity-preserving maps in a way that has implications to the choice of scalings and transformations in the derivation of optimization algorithms in general. The third purpose of this note is to point this out.