32 I. Convex Sets at Large (7.6) Corollary. Let A1,... , Am Rd be sets such that [Ai] K(Rd) for all i = 1, . . . , m. Then [A1 . . . Am] K(Rd) and χ(A1 . . . Am) = m k=1 (−1)k−1 1≤i1i2...ik≤m χ(Ai 1 . . . Ai k ). In particular, χ(A1 A2) = χ(A1) + χ(A2) χ(A1 A2). Proof. Follows by Lemma 7.2 and Theorem 7.4. PROBLEMS. 1. Let A1,A2,A3 Rd be closed convex sets such that A1∩A2 = ∅, A1∩A3 = ∅, A2 A3 = and A1 A2 A3 is a convex set. Prove that A1 A2 A3 = ∅. 2. Let A1,... , Am Rd be closed convex sets such that A1 . . . Am is a convex set. Suppose that the intersection of every k sets Ai 1 , . . . , Ai k is non-empty. Prove that there are k + 1 sets Ai 1 , . . . , Ai k+1 whose intersection is non-empty. 3. Let Δ = (ξ1, . . . , ξd) : ξ1 + . . . + ξd = 1, ξi 0 for i = 1, . . . , d be the standard simplex in Rd. Let Δi = x Δ : ξi = 0 be the i-th facet of Δ. Suppose that there are compact convex sets K1,... , Kd Rd, such that Δ K1 . . . Kd and Ki Δi = for i = 1, . . . , d. Prove that K1 . . . Kd = ∅. Hint: Use induction on d and Problem 2. 4. Let A1,... , Am Rd be closed convex sets such that A1 . . . Am = ∅. Prove that χ(A1 . . . Am) = 1. 5. Find the Euler characteristic of the “open square” I2 = (ξ1, ξ2) : 0 ξ1,ξ2 1 and the “open cube” I3 = (ξ1, ξ2,ξ3) : 0 ξ1,ξ2,ξ3 1 . 6. Let A1,A2,A3,A4 Rd be closed convex sets such that the union A1 A2 A3 A4 is convex and all pairwise intersections A1 A2,A1 A3,A1 A4,A2 A3,A2 A4 and A3 A4 are non-empty. Prove that at least three of the four intersections A1 A2 A3,A1 A2 A4,A1 A3 A4 and A2 A3 A4 are non- empty and that if all the four intersections are non-empty, then the intersection A1 A2 A3 A4 is non-empty. Construct an example where exactly three of the four intersections A1 A2 A3,A1 A2 A4,A1 A3 A4 and A2 A3 A4 are non-empty.
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