Isosceles triple convexity

Description

A set $S$ in $\mathbb{R}^{d}$ is called $it$ -convex if, for any two distinct points in $S$ , there exists a third point in $S$ , such that one of the three points is equidistant from the others. In this paper we first investigate nondiscrete $it$ -convex sets, then discuss about the $it$ -convexity of the eleven Archimedean tilings, and treat subsequently finite subsets of the square lattice. Finally, we obtain a lower bound on the number of isosceles triples contained in an $n$ -point $it$ -convex set.