Projection of a convex set on a subspace

The projection of a point on a set is the point that is closest (in the sense of Euclidean norm) to the set. Mathematically the projection of a point on a convex set {cal C} is the (unique) solution to the optimization problem

$pi(x) := argmin_y : |x-y|_2 ~:~ y in {cal C}.$

The projection of a whole set on another set is simply the set of projections pi(x) , when runs {cal C} .

The projection of a convex set on any subspace or affine set is convex.

The image shows the convex set

$left{ (x,y,z) ~: y ge x^2, ;; z ge y^2 right}$

and its projection on the space of (x,y) variables. The projection turns out to be the set

$left{ (x,y) ~: y ge x^2 right}.$

Since the original set is convex, its projection also is.