Convex geometry

Convex geometry studies sets that contain the line segment between any two of their points, along with their geometric properties and relations.

Convex set: A set is convex if it contains the line segment joining every pair of its points. This defining property is the starting point for the subject.
Separation theorem: A theorem that separates disjoint convex sets by a hyperplane under suitable conditions. It connects convexity to linear inequalities and boundary geometry.
Convex optimization: Optimization of a convex objective over a convex feasible region. Convex sets make local optima global under standard conditions.
Differential geometry: The study of smooth manifolds and geometric structures using differential calculus. It emphasizes curvature and local smooth structure rather than convexity alone.
Convex hull: The smallest convex set containing a given set of points. It turns arbitrary point configurations into the convex sets studied here.
Krein–Milman theorem: A theorem stating that a compact convex set in a locally convex space is the closed convex hull of its extreme points. It shows how extreme points determine an entire compact convex set.
Linear programming: Optimization of a linear objective subject to linear equality and inequality constraints. Its feasible regions are polyhedra, a central class of convex sets.
Projective geometry: The study of geometric properties invariant under projective transformations. Its incidence-based viewpoint differs from convexity's focus on segments and containment.
Supporting hyperplane: A hyperplane that meets a convex set while leaving the entire set in one of its closed half-spaces. Supporting hyperplanes expose boundary structure and help characterize convex sets.
Carathéodory's theorem: A theorem stating that points in a convex hull in dimension d need at most d+1 generating points. It bounds how many points are needed to represent a point in a convex hull.

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