(related to Corollary: Intersection of Convex Affine Sets)
Let \mathcal C be a family of convex subsets C of an n-dimensional affine space \mathcal A=(A,V_A,v) with V_A as the associated vector space over the field \mathbb R of real numbers. Take two points P,Q from the intersection \bigcap\mathcal C.
By definition of set intersection, for all C\in\mathcal C we have that P,Q\in C.
The convexity of each C yields that the straight line \lambda P+(1-\lambda)Q is contained in each C.
Therefore, these elements are also in the intersection \bigcap\mathcal C, by definition of set intersection.
Hence \bigcap\mathcal C is also convex.
