Klee-Phelps Convex Groupoids

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 $N_z(\circ )=S_z(\circ )$.