Analysis II | Part IB, 2005

Use the standard metric on Rn\mathbf{R}^{n} in this question.

(i) Let AA be a nonempty closed subset of Rn\mathbf{R}^{n} and yy a point in Rn\mathbf{R}^{n}. Show that there is a point xAx \in A which minimizes the distance to yy, in the sense that d(x,y)d(a,y)d(x, y) \leqslant d(a, y) for all aAa \in A.

(ii) Suppose that the set AA in part (i) is convex, meaning that AA contains the line segment between any two of its points. Show that point xAx \in A described in part (i) is unique.

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