12 I. Convex Sets at Large

(2.4) Corollary. If S ⊂

Rd

is a compact set, then conv(S) is a compact set.

Proof. Let Δ ⊂ Rd+1 be the standard d-dimensional simplex; see Problem 1 of

Section 2.2:

Δ = (α1, . . . , αd+1) :

d+1

i=1

αi = 1 and αi ≥ 0 for i = 1, . . . , d + 1 .

Then Δ is compact and so is the direct product

Sd+1

×Δ =

(

u1,... , ud+1; α1,... , αd+1

)

: ui ∈ S and (α1, . . . , αd+1) ∈ Δ .

Let us consider the map Φ : Sd+1 × Δ −→ Rd,

Φ(u1, . . . , ud+1; α1,... , αd+1) = α1u1 + . . . + αd+1ud+1.

Theorem 2.3 implies that the image of Φ is conv(S). Since Φ is continuous, the

image of Φ is compact, which completes the proof.

PROBLEMS.

1. Give an example of a closed set in R2 whose convex hull is not closed.

2. Prove that the convex hull of an open set in Rd is open.

3. An Application: Positive Polynomials

In this section, we demonstrate a somewhat unexpected application of Cara-

th´ eodory’s Theorem (Theorem 2.3). We will use Carath´ eodory’s Theorem in the

space of (homogeneous) polynomials.

Let us fix positive integers k and n and let H2k,n be the real vector space of

all homogeneous polynomials p(x) of degree 2k in n real variables x = (ξ1, . . . , ξn).

We choose a basis of H2k,n consisting of the monomials

ea = ξ1

α1

· · · ξn

αn

for a = (α1, . . . , αn) where α1 + . . . + αn = 2k.

Hence dim H2k,n =

(

n+2k−1

2k

)

. At this point, we are not particularly concerned

with choosing the “correct” scalar product in H2k,n. Instead, we declare {ea} the

orthonormal basis of H2k,n, hence identifying H2k,n = Rd with d =

(

n+2k−1

2k

)

.

We can change variables in polynomials.

(3.1) Definition. Let U :

Rn

−→

Rn

be an orthogonal transformation and let

p ∈ H2k,n be a polynomial. We define q = U(p) by

q(x) = p

(

U

−1

x

)

for x = (ξ1, . . . , ξn).

Clearly, q is a homogeneous polynomial of degree 2k in ξ1,... , ξn.