Projection: The projection of a vector on a set is a vector in the set closest to the vector.
Distance: The distance of a vector to a set is its distance to its projection on the set.
Closed convex set (Dual definition): A set is closed and convex, if it is the intersection of all closed half-spaces containing the set.
Global & local minima: Convex function has identical local and global minima.
{Bertsekas2003, 2.2.1} Projection theorem:
For a (nonempty) closed convex set C,
x on C exists and is unique.C relative to which the orthogonal hyperplane of x separates C and x.C is a continuous and non-expansive map (weak contractor).C is a convex function.{Bertsekas2003, 2.4.1} Supporting Hyperplane Theorem: Any non-interior point of a (nonempty) convex set has some hyperplane that contains the set in one of its closed half-spaces.
[If a set is closed with nonempty interior and has a supporting hyperplane at every point in its boundary, it is convex.]
{Bertsekas2003, 2.4.2} Separating Hyperplane Theorem: Any two disjoint (nonempty) convex sets can be separated by some hyperplane.
{Bertsekas2003, 2.4.3} Strict Separation Theorem: Two disjoint (nonempty) convex sets can be strictly separated by some hyperplane if any of the following holds:
C_1 - C2]