 Closest point theorem for a closed convex subset of Lp

For $2\le p<\infty$, a nonempty closed convex set $K$ in an Lp space has exactly one nearest point to every $f$. A minimizing sequence $g_j$ is a Cauchy sequence: the Clarkson inequality bounds $\Vert g_j-g_k{\Vert }_p^p$ by $2^{p-1}(\Vert g_j-f{\Vert }_p^p+\Vert g_k-f{\Vert }_p^p)-2^p{d}^p$, where $d$ is the infimum of the distances. Completeness of Lp spaces and the closed set property give a limit in $K$. Applying the same inequality to two minimizers proves uniqueness. Nonemptiness is essential.