Convex function

A function whose value at any point on a line segment is no greater than the linear interpolation of its values at the segment’s endpoints.

Convex set: A set containing the entire line segment between any two of its points. Convex functions are defined by testing line segments within convex domains.
Monotone derivative: A derivative that never decreases as its input increases. For a differentiable function on an interval, this is equivalent to convexity.
Convex optimization: Optimization of a convex objective over a convex feasible set. Every local minimum of a convex objective over a convex domain is globally optimal.
Concave function: A function whose value along a segment is at least the linear interpolation of endpoint values. It reverses the defining inequality and corresponds to convexity after negation.
Johan Jensen: A Danish mathematician known for Jensen's inequality and work in analysis. His inequality gave convexity a powerful formulation for averages and expectations.
Secant line: A line passing through two points on a curve. The convexity inequality places the graph below every chord joining two graph points.
Hessian matrix: The matrix of second partial derivatives of a scalar-valued function. Positive semidefiniteness of the Hessian gives a multivariable convexity test.
Exponential family: A class of probability distributions whose densities have a standard exponential form. Their log-partition functions are convex, enabling estimation and uncertainty analysis.
Quasiconvex function: A function whose sublevel sets are convex. Quasiconvexity constrains level sets but does not require the full chord inequality.
Hermann Minkowski: A German mathematician whose work shaped geometry of numbers and convex geometry. His geometric methods helped establish convex sets as a major mathematical subject.

Linked from 22 pages