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Jung's theorem states that any subset of $\mathbb{R}^n$ with unit diameter is contained in some ball with radius equal to $\sqrt{\frac{n}{2(n+1)}}$.

I'm interested in more general

Problem: For $n,m\in\mathbb{N}$ find the smallest constant $r(n,m)$ such that any subset of $\mathbb{R}^n$ with unit diameter can be covered by collection of $m$ balls, each with radius $r(n,m)$.

Do you know any results dealing with the above?

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