22 I. Convex Sets at Large

is non-empty. Since the sets Ar and Ab are open, their intersection is an open set;

therefore an intersection of sets Ar and Ab is non-empty if and only if it contains

a point (c, α) with c = 0. Hence for any subset S ⊂ R ∪ B, the intersection

r∈S∩R

Ar ∩

b∈S∩B

Ab

is not empty if and only if there is a point (c, α) ∈ Rd+1 such that the hyperplane

H = x ∈

Rd

: c, x = α

strictly separates the sets B ∩ S and R ∩ S. This completes the proof.

For example, if two sets of points in the plain cannot be separated by a straight

line, one of the three configurations of Figure 6 must occur.

1 ) 2 ) 3 )

Figure 6. The three reasons points cannot be separated in the plane

PROBLEMS.

1. Prove that if a convex set is contained in the union of a finite family of

halfspaces in Rd (sometimes we say covered by a finite family of halfspaces; see

Section 5.2), then it is contained in the union of some d + 1 (or fewer) halfspaces

from the family (covered by some d + 1 subspaces).

2. Let I1,... , Im be parallel line segments in R2, such that for every three

Ii1 , Ii2 , Ii3 there is a straight line that intersects all three. Prove that there is a

straight line that intersects all the segments I1,... , Im.

3. Let Ai : i = 1, . . . , m be convex sets in R2 such that for every two sets Ai

and Aj there is a line parallel to the x-axis which intersects them both. Prove that

there is a line parallel to the x-axis which intersects all the sets Ai.

(5.2) The center point. Let us fix a Borel probability measure μ on

Rd.

This

means, roughly speaking, that for any “reasonable” subset A ⊂

Rd

a non-negative

number μ(A) is assigned which satisfies some additivity and continuity properties

and such that

μ(Rd)

= 1. We are not interested in rigorous definitions here; the

following two examples are already of interest: