= Solution
The function $\phi(x)=\max_{v\in C}\langle x,v\rangle$ is the <support function> $\sigma_C$. For a nonempty compact <convex set>,
$$
\sigma_C^*(v)=
\begin{cases}
0,&v\in C,\\
+\infty,&v\notin C,
\end{cases}
$$
so its <convex conjugate> is the <indicator function> $\iota_C$. Applying the <Moreau decomposition>,
$$
\operatorname{prox}_{t\phi}(y)
=y-t\operatorname{prox}_{t^{-1}\iota_C}(y/t).
$$
Multiplication of an <indicator function> by a positive scalar does not change it, and its <proximal operator> is the <Euclidean projection onto a convex set>. Therefore
$$
\boxed{\operatorname{prox}_{t\phi}(y)=y-tP_C(y/t)}.
$$
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