Convex set

A subset of a real vector space is convex if, for every pair of its points, it contains the entire line segment joining them.

Line segment: The points between two endpoints, including the endpoints, on a straight line. Convexity requires every segment joining two points of the set to stay inside it.
Convex hull: The smallest convex set containing a given set. It is formed by adding every convex combination of the original points.
Convex optimization: Optimization of a convex objective over a convex feasible region. Convex feasible sets help ensure that local optima are global optima.
Star-shaped set: A set containing at least one point from which every point in the set is visible along a segment. Unlike convexity, visibility from a single point does not require every pair's segment.
Convex combination: A weighted sum of points whose weights are nonnegative and sum to one. Every point on a segment is a convex combination of its endpoints.
Intersection of convex sets: The intersection of any collection of convex sets is convex. A point on a segment in every set remains in their shared intersection.
Linear programming: Optimization of a linear objective subject to linear inequalities and equalities. Its feasible regions are polyhedra, a central class of convex sets.
Connected set: A topological space that cannot be split into two disjoint nonempty open subsets. Convex sets are connected, but connected sets need not contain segments between their points.
Affine space: A geometric space in which points can be combined using coefficients that sum to one. Its affine structure defines segments without requiring a chosen origin.
Minkowski sum: The set of all sums formed by adding one point from each of two sets. Adding points from convex sets preserves convexity of the resulting set.

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