Extreme point

An extreme point of a convex set is a point that cannot be written as a convex combination of two distinct points in that set, with both coefficients positive.

Convex set: A set is convex if it contains the line segment between every pair of its points. Extreme points are defined relative to a convex set and its line segments.
Exposed point: An extreme point that uniquely maximizes some continuous linear functional on a convex set. Supporting hyperplanes identify a particularly visible subclass of extreme points.
Linear programming: Optimization of a linear objective over a region defined by linear constraints. When an optimum exists on a polytope, some extreme point attains it.
Vertex (geometry): A corner point of a geometric figure, commonly defined through its incident edges or faces. For polytopes, vertices coincide with extreme points, but the terms arise from different viewpoints.
Convex combination: A weighted sum of points with nonnegative coefficients that sum to one. A point is extreme precisely when no nontrivial such combination of other points produces it.
Supporting hyperplane: A hyperplane meeting a convex set while the set lies entirely in one of its closed half-spaces. A supporting hyperplane can expose an extreme point as the unique maximizing point.
Polytope: A bounded intersection of finitely many half-spaces in finite-dimensional Euclidean space. A polytope has finitely many extreme points, which encode its vertices.
Boundary point: A point every neighborhood of which meets both a set and its complement. Boundary membership alone does not prevent a point from lying inside a segment in the set.
Affine subspace: A translate of a linear subspace, containing every affine combination of its points. Extreme-point arguments concern affine combinations, which preserve affine subspaces.
Krein–Milman theorem: A compact convex subset of a locally convex space is the closed convex hull of its extreme points. The theorem makes extreme points sufficient to reconstruct compact convex sets.

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