We consider projection on arbitrary convex sets in finite-dimensional Euclidean space; convex because projection is, then, unique minimum-distance and a convex optimization problem:

If C is a closed convex set,
then for each and every x there exists a unique point Px
belonging to C that is closest to x in the Euclidean sense.
Unique projection Px of
a point x on convex set C
is that point in C closest to x.
There exists a converse:

If C is a nonempty closed set
and if for each and every x there is
a unique Euclidean projection Px of x on C
belonging to C, then C is convex.

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