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The convex hull of a nonempty set consists of its finite convex combinations.
Theorem. Let be a nonempty subset of a real vector space . Then the convex hull can be described as
Let be the set of all finite convex combinations of points in . By closure under convex combinations, is convex and contains , hence by minimality. Conversely, every convex set containing contains all such combinations, so .
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.
Proposition. A nonempty set in a real vector space is convex if and only if it contains every convex combination of finitely many of its points; i.e., for all , all , and all with , we have
Context. This is the "finite averaging" characterization of convexity and is used to describe convex hulls and many convex constructions.
Proof sketch. The "only if" direction follows by induction on using the two-point definition of convexity. For "if," take to recover the defining two-point condition.