Let $H$ be a Hilbert space and $S,K\subseteq H$ be two bounded closed convex sets such that $S\subseteq K$. Denote by $\partial K, \partial S$ the boundary of $K$ and $S$ respectively. Let $P_S:H\to S$ denote the metric projection on $S$. Then it is not difficult to show that $P_S(\partial K)\subseteq \partial S$. Is the reverse inclusion $\partial S\subseteq P_S(\partial K)$ true?
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