Convex hull

The convex hull of a set is the set of all finite convex combinations of its points, equivalently the smallest convex set containing it.

The smallest closed convex set containing a set $C$, equivalently the closure of its convex hull in the specified topology. The topology must be stated: the norm topology, weak topology and weak-star topology need not give the same closure for an arbitrary set.

Every point in the convex hull of a subset of ${\mathbb{R}}^d$ is a convex combination of at most $d+1$ points of that subset.

In an $N$-dimensional real vector space, each member of a conic hull has a representation using at most $N$ generators. To prove it, take a finite positive-coefficient representation with more than $N$ terms. Its generators are linearly dependent, say ${\sum }_j{\mu }_j{s}_j=0$, with some ${\mu }_j>0$ after reversing the relation if necessary. Subtract $t{\mu }_j$ from each coefficient, taking $t={\min }_{{\mu }_j>0}{\lambda }_j/{\mu }_j$. All coefficients stay nonnegative and at least one vanishes. Iterate. The bound differs from the $N+1$ bound for a convex hull because the coefficient sum is unrestricted.