 Solution

ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2010/p/p3/c/solution

Suppose

u, v \in U_{n}

both attain the minimum distance

d

. Apply (b) to

x - u

and

x - v

∥ u - v ∥^{2} + 4 x - \frac{u + v}{2}^{2} = 2∥ x - u ∥^{2} + 2∥ x - v ∥^{2} = 4 d^{2} .

The midpoint belongs to

U_{n}

, so its squared distance is at least

d^{2}

. Thus

∥ u - v ∥^{2} \leq 0

u = v

. Existence will also follow constructively in (d)(i).

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