Page "Convex hull" Paragraph 9

It is not obvious that the first definition makes sense: why should there exist a unique minimal convex set containing X, for every X?

However, the second definition, the intersection of all convex sets containing X is well-defined, and it is a subset of every other convex set Y that contains X, because Y is included among the sets being intersected.

Thus, it is exactly the unique minimal convex set containing X.

Each convex set containing X must ( by the assumption that it is convex ) contain all convex combinations of points in X, so the set of all convex combinations is contained in the intersection of all convex sets containing X. Conversely, the set of all convex combinations is itself a convex set containing X, so it also contains the intersection of all convex sets containing X, and therefore the sets given by these two definitions must be equal.