Klee-Phelps Convex Groupoids

TL;DR

This paper investigates the properties of proximal Klee-Phelps convex groupoids in finite-dimensional normed spaces, establishing conditions for normed proximality and convexity of groupoid neighborhoods.

Contribution

It proves the equivalence of proximality and normed proximality for Klee-Phelps convex groupoids and characterizes convexity of groupoid neighborhoods.

Findings

Proximal Klee-Phelps convex groupoids are normed proximal.

Groupoid neighborhoods are convex if and only if they coincide with their supersets.

The paper provides conditions for convexity in finite-dimensional normed spaces.

Abstract

We prove that a pair of proximal Klee-Phelps convex groupoids $A(\circ)$,$B(\circ)$ in a finite-dimensional normed linear space $E$ are normed proximal, {\em i.e.}, $A(\circ)\ \delta\ B(\circ)$ if and only if the groupoids are normed proximal. In addition, we prove that the groupoid neighbourhood $N_z(\circ)\subseteq S_z(\circ)$ is convex in $E$ if and only if $N_z(\circ) = S_z(\circ)$.