On $r$-separated sets in normed spaces

Abstract:

The separation of a bounded set $A$ in a metric space $\delta (A)$ is defined as the supremum of the numbers $r > 0$ such that there exists a sequence $({x_n})$ in $A$ such that $d({x_n},{x_m}) > r$ for every $n \ne m$. We prove for every bounded set $A$ in a Banach space that $\delta (A) = \delta ({\text {co}}(A))$ where ${\text {co}}(A)$ denotes the convex hull of $A$. This yields a generalization of Darbo’s fixed point theorem.