How do you prove that the convex hull of A is the smallest convex set containing A?
edit: definition of a convex hull: Given a set A ⊆ ℝn the set of all convex combinations of points from A is called the convex hull of A.
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Young, when you wrote
How do you prove that the convex hull of A is the smallest set containing A?
You meant that convex hull of A is the minimal convex set containing A, right?
To show this, which part is your definition? The linear-algebraic characterization?
You can see that any intersection of convex sets containing A is also a convex set containing A.