extremepoints

An extremepoint is a concept in convex geometry and optimization. A point in a convex set is called an extremepoint if it cannot be expressed as a convex combination of two distinct points in the set. In simpler terms, an extremepoint is a "corner" or "vertex" of the convex set.

For a convex set K, a point x belonging to K is an extremepoint if for any two points y and z in K and any scalar t between 0 and 1 (exclusive of 0 and 1), if x = ty + (1-t)z, then x must be equal to y and x must be equal to z. This means that an extremepoint is not on any line segment connecting two other points within the set, except for the degenerate case where the segment consists only of the extremepoint itself.

The set of extremepoints of a convex set can be used to characterize the set. For instance, the Krein-Milman theorem states that a compact convex set in a locally convex topological vector space is the closed convex hull of its extremepoints. This means that the entire set can be "reconstructed" from its corners.

Extremepoints are particularly important in linear programming, where the optimal solution to a linear program over a polyhedron (a type of convex set) always occurs at an extremepoint. Identifying and evaluating these points is a fundamental strategy in solving such problems. The concept also appears in functional analysis and other areas of mathematics.