Orthogonal projection of a vector onto convex set

Question

What is the meaning of an orthogonal projection of a vector onto a convex set? I am familiar with orthogonal projection of a vector onto a vector space but I cannot imagine how it works with a set. What is the projection looks like?

Answer 1

I assume that we are working on a real inner product space. The orthogonal projection of a vector $v$ on a convex set $C$ is a vector $v^\star\in C$ such that, for each $w\in C$,$$\bigl\langle v-v^\star,w-v^\star\bigr\rangle\leqslant0.$$It can be proved that, if the space is a Hilbert space and if $C$ is not only convex but also closed, then, for each $v$, $v^\star$ exists and it is unique.