# Orthogonal projection of a vector onto convex set

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?

• It probably refers to the nearest point in the convex set. That is, given $x$ in the ambient space, $P(x)$ is the point in the convex set that minimizes the distance $\|x-c\|$, where $c$ runs in the convex set. – Giuseppe Negro Aug 31 '18 at 15:48

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.
• That's a way of seeing it. The plane passing through $v^\star$ which is orthogonal to $v-v^\star$ divides the whole space into two half-spaces. That inequality means that $v$ belongs to one of them, whereas the whole of $C$ is a subset of the other one. – José Carlos Santos Aug 31 '18 at 19:07