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Any intersection of convex sets is convex
Proposition. Let be a family of convex subsets of a vector space . Then the intersection is convex.
Context. This is fundamental for defining the convex hull as an intersection of all convex supersets and for building convex feasible regions from many convex constraints.
Proof sketch. If lie in every , then for each and each the point lies in by convexity. Hence it lies in the intersection.
Let be a vector space (typically over ). A set is convex if for all and all we have
Context. Equivalently, is convex iff it contains every line segment joining any two of its points. Convexity is the core geometric notion underlying convex analysis and optimization.
Let be a real vector space and let . The convex hull of , denoted , is defined as the intersection of all convex sets containing :
Context. By stability under intersections, this definition ensures is convex. The convex hull is the basic "convexification" operator.