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).
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