Criteria for approximation of linear and affine functions
We consider a subset S of \({\mathfrak R}^ m\) and we suppose that to each \(x\in S\) is associated a convex subset C(x) of \({\mathfrak R}^ k\); we give then a criterion for the existence of a linear or affine function \(f: {\mathfrak R}^ m\to {\mathfrak R}^ k\) such that for every \(x\in S\), f(x)\(\in C(x)\). This contains as particular cases Helly's Theorems, the Transversal Theorem of Santaló, and some results of Grünbaum and Karlin-Shapley. We concentrate then on the case \(k=1\) and derive criteria for a function \(g: S\to {\mathfrak Z}\) to be the approximation of a linear or affine function \(f: S\to {\mathfrak R}\). We show how this is related to the definition of straightness in digital geometry.
scientific article; zbMATH DE number 3903452
scientific article