Let $\Omega\subset \mathbb R^d$ be an open, convex domain, and consider the Hilbert space $L^2(\Omega)$. Each sum of convex functions is convex, hence the subset $Conv(\Omega)$ of all convex functions is convex. I also have the strong impression that $Conv(\Omega)$ is closed wrt the $L^2$-norm, in spite of the fact that its elements are continuous. So, there exists an orthogonal projector of $L^2(\Omega)$ onto $Conv(\Omega)$. Can one say how it looks like?

Let $\Omega\subset \mathbb R^d$ be an open, convex domain, and consider the Hilbert space $L^2(\Omega)$. Each sum of convex functions is convex, hence the subset $Conv(\Omega)$ of all convex functions is convex. I also have the strong impression that $Conv(\Omega)$ is closed wrt the $L^2$-norm, in spite of the fact that its elements are continuous. So, there exists a projector of $L^2(\Omega)$ onto $Conv(\Omega)$. Can one say how it looks like?