6 I. Convex Sets at Large

In particular, if B = {b} is a point, the set

A + b = x + b : x ∈ A

is a translation of A. For a number α and a subset X ⊂ V , the set

αX = αx : x ∈ X

is called a scaling of X (for α 0, the set αX is also called a dilation of X). Some

properties of convex sets are obvious, some are not so obvious, and some are quite

surprising.

PROBLEMS.

We will encounter some of the harder problems below later in the text.

1◦.

Prove that the intersection

i∈I

Ai of convex sets is convex.

2◦.

Let A ⊂ V be a convex set and let T : V −→ W be a linear transformation.

Prove that the image T (A) is a convex set in W .

3. Let A ⊂

Rn

be a polyhedron (see Problem 2, Section 1.3) and let T :

Rn

−→

Rm

be a linear transformation. Prove that the image T (A) is a polyhedron in

Rm.

Remark: We prove this in Section 9; see Theorem 9.2.

4. Prove that A + B is a convex set provided A and B are convex. Prove that

for a convex set A and non-negative numbers α and β one has (α+β)A = αA+βA.

Show that the identity does not hold if A is not convex or if α or β are allowed to

be negative.

5∗.

For a set A ⊂

Rd,

let [A] :

Rd

−→ R be the indicator function of A:

[A](x) =

1 if x ∈ A,

0 if x / ∈ A.

Let A1,... , Ak be compact convex sets in

Rn

and let T :

Rn

−→

Rm

be a linear

transformation. Let Bi = T (Ai) be the image of Ai. Suppose that

∑k

i=1

αi[Ai] = 0

for some numbers αi. Prove that

∑k

i=1

αi[Bi] = 0. Show that this is no longer true

if the Ai are not convex.

Remark: We prove this in Section 8; see Corollary 8.2.

6. Let A ⊂

Rd

be a compact convex set and B = (−1/d)A. Prove that there

exists a vector b ∈ Rd such that b + B ⊂ A.

Remark: Figure 2 illustrates the statement for d = 2. We go back to this

problem in Section 5 when we discuss Helly’s Theorem (see Problem 1 of Section

5.2 and the hint thereafter).