Prove that the intersection of two convex sets is convex. Show by example that the union of two convex sets need not to be convex. Is the empty set convex?
(a) By definition a set is convex if for any points P and Q in the set, the segment `bar(PQ)` is also in the set.
Consider the set `L=L_1 nn L_2` where `L_1,L_2` are convex.
Take `P,Q in L` . By definition `P,Q in L_1` and since `L_1` is convex `bar(PQ) in L_1` . Simarly `bar(PQ) in L_2` so `bar(PQ) in L_1 nn L_2` which implies that L is convex.
** If the intersection consists of 1 point L is convex since P and Q are the same point so the "segment" is in L. If the intersection is empty then L is convex since the empty set is convex. **
(b) Consider the set `L_1` in Euclidean 2-space -- it is the line segment with endpoints (0,0) and (2,2). Let `L_2` be a line segment in Euclidean 2-space with endpoints (2,2) and (4,0).
Both `L_1,L_2` are convex. Also `(1,1) in L_1 uu L_2,(3,1) in L_1 uu L_2` . But only the endpoints of the segment joining (1,1) and (3,1) are in the union, so the union is not convex.