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A convex set in light blue, and its extreme points in red.

An extreme point or an extremal point is a point that belongs to the extremity of something.

• In mathematics, an extreme point of a convex set S in a real vector space is a point in S which does not lie in any open line segment joining two points of S. Intuitively, an extreme point is a "corner" of S. The Krein–Milman theorem is one of the most well-known applications of extreme points. It says that if S is convex and compact in a locally convex space, then S is the convex hull of its extreme points.
• In mathematics, the term should not be confused with a similar notion of extremal point which is a point where some function attains its extremum. For example, points in the plane that have minimal or maximal X-coordinates are called extremal points in X direction. Similarly extremal points may be defined for any direction, not only in a direction of a coordinate axis.
• In graph theory, the leaf vertices of a tree are sometimes called extremal points or extremal vertices.
• In geography, an extreme (extremal) point is a point of land that extends farther in one direction than any other part of that land, which corresponds to the mathematical notion of extremal point in a given direction. See Extreme points of the world.

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