Gruber P. Convex and Discrete Geometry

44 Convex Bodies
The next result is a simple consequence of Carathéodory’s theorem.
Corollary 3.1. Let A ⊆ E d be compact. Then conv A is compact.
Proof. The set
� (λ1,...,λd+1, x1,...,xd+1) : λi ≥ 0,λ1 +···+λd+1 = 1, x j ∈ A �
is a compact subset of Ed+1+(d+1)d = E (d+1)2 . Hence its image under the continuous
mapping
of E (d+1)2
(λ1,...,λd+1, x1,...,xd+1) → λ1x1 +···+λd+1xd+1
into E d is also compact. By Carathéodory’s theorem this image is conv A.
⊓⊔
Closure and Interior
In the following some useful minor results on closure and interior of convex sets and
on convex hulls of subsets of E d are presented. Let C ⊆ E d be convex. By relint C
we mean the interior of C relative to the affine hull aff C of C, i.e. the smallest flat in
E d containing C.
Proposition 3.1. Let C ⊆ E d be convex. Then the following statements hold:
(i) cl C is convex.
(ii) relint C is convex.
(iii) C ⊆ cl relint C.
Let B d denote the solid Euclidean unit ball in E d .
Proof. (i) To show that cl C is convex, let x, y ∈ cl C and 0 ≤ λ ≤ 1. Choose
sequences (xn), (yn) in C such that xn → x, yn → y as n →∞. The convexity of
C implies that (1 − λ)xn + λyn ∈ C. Since (1 − λ)xn + λyn → (1 − λ)x + λy, it
follows that (1 − λ)x + λy ∈ cl C.
(ii) It is sufficient to consider the case where int C �= ∅and to show that int C is
convex. Let x, y ∈ int C and 0 ≤ λ ≤ 1. Choose δ>0 such that x +δB d , y +δB d ⊆
int C ⊆ C. Then
(1 − λ)x + λy + δB d = (1 − λ)(x + δB d ) + λ(y + δB d ) ⊆ C
by the convexity of B d and C. Thus (1 − λ)x + λy ∈ int C.
(iii) It is sufficient to consider the case where int C �= ∅and to prove that C ⊆
cl int C.Letx ∈ C and choose y ∈ int C. Then there is a δ>0 such that y + δB d ⊆
int C ⊆ C. The convexity of C now implies that
(1 − λ)x + λy + λδB d = (1 − λ)x + λ(y + δB d ) ⊆ C.
Thus (1 − λ)x + λy ∈ int C for 0