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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