Convexity

Convexity is a property of sets that contain every line segment joining two of their points, and of functions whose values satisfy a corresponding inequality. It underlies geometric containment and optimization.

Convex set: A set containing every line segment between any two of its points. This is the basic set-theoretic definition of geometric convexity.
Second derivative test for convexity: A criterion that uses nonnegative second derivatives to establish convexity on an interval. For twice-differentiable functions, curvature gives a local test for global convexity.
Convex optimization: Optimization of a convex objective over a convex feasible set. Convexity ensures that every local minimum is global.
Convex hull: The smallest convex set containing a given set of points. It completes any set into the smallest set satisfying convexity.
Concave function: A function whose value at a weighted average is no less than the weighted average of its values. Concavity reverses the defining inequality for convex functions.
Convex function: A function whose value at a weighted average is no greater than the same weighted average of its values. This is the functional version of convexity, defined by an inequality.
Hessian matrix: The matrix of second-order partial derivatives of a scalar-valued function. A positive semidefinite Hessian establishes convexity on a convex domain.
Linear programming: Optimization of a linear objective subject to linear equality and inequality constraints. Its feasible region is a convex polyhedron, making it a foundational convex optimization problem.
Convex geometry: The study of convex sets and their geometric properties. This field develops the geometric side of convexity.
Nonconvex optimization: Optimization in which the objective or feasible region is nonconvex. Without convexity, local minima may fail to be global and solutions can be harder to characterize.

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