TITLE: a set of diameter $d$ is contained in a ball of diameter $d$?
QUESTION [7 upvotes]: Suppose $S$ is a connected open subset of $\mathbb{R}^n$, and $d$ is the diameter of $S$. Is $S$ contained in some ball of radius $d/2$?

REPLY [8 votes]: I think as you say, a solid equilateral triangle with side lengths $d$ disproves this.  The obvious choice of centre for the ball would be the centroid but the distances to the diagonal are greater than $\frac{d}{2}$.