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Can anyone please explain to me how one could apply the nearest theorem to deduce if there is a unique point, nearest to a set.

E.g., Show that if $F$ is a non-empty closed set in ${\bf R}^p$ and if $x$ is not an element of $F$, there is a unique point of $F$ that is nearest to $x$.

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    You write that you want to apply "the nearest theorem." I assume you want to apply "the nearest point theorem." If that's right, can you edit your question accordingly, and also state the nearest point theorem, so we know what we are trying to apply?2012-05-07

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