What is real $R$ so that every subset of Euclidean space with diameter one is inside a ball of radius $R$?

Question

What is infimum of real numbers $R$ so that for every $n$ every $S \subseteq \mathbb{E}^n$, for which $d(S) = \sup\{|x-x'| \mid x,x' \in S \} = 1$, is inside some closed $n$-ball of radius $R$? In particular, is this infimum $\sqrt{2}/2$ and is in fact every such $S$ inside closed ball of radius strictly less than $\sqrt{2}/2$?

Answer 1

Jung's Theorem gives the definitive answer to this question: Given any compact set $K\subset{\mathbb R}^n$ with ${\rm diam}(K)\leq1$ there is a unique ball $B_r\supset K$ of minimal radius, and $$r\leq\sqrt{{n\over 2(n+1)}}\ .$$