What minimizes the Chebyshev Distance?

Question

For an arbitrary number of dimensions, I know that the mean minimizes the distance using the $L_2$ norm and that the geometric median minimizes the distance function using the $L_1$ norm (though I have yet to find a good proof of this). So what minimizes the $L_{\infty}$ (or Chebyshev) norm?

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Answer 1

The mid-range minimizes the $\:L_{\hspace{.02 in}\infty}\:$ norm.