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(Ai1 ∩ . . . ∩ Aik ).

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 Ai1 , . . . , Aik is non-empty.

Prove that there are k + 1 sets Ai1 , . . . , Aik+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.