Gruber P. Convex and Discrete Geometry

4 Support and Separation 59
(i)⇒(ii) For the proof of the convexity of C − D, let x, z ∈ C, y,w ∈ D and
0 ≤ λ ≤ 1. Then
(1 − λ)(x − y) + λ(z − w) = � (1 − λ)x + λz � − � (1 − λ)y + λw � ∈ C − D
by the convexity of C and D. Thus C − D is convex. Next, let S ={x : α ≤ u · x ≤
β},α 0, separate {o} and
C − D. Then u · (x − y) ≤−γ , i.e. u · x + γ ≤ u · y for all x ∈ C and y ∈ D.
Let α = sup{u · x : x ∈ C}. Then the slab {z : α ≤ u · z ≤ α + γ } separates C
and D. ⊓⊔
Theorem 4.4. Let C, D ⊆ E d be convex. Then the following hold:
(i) Let C be compact, D closed and C ∩ D = ∅. Then C and D are strongly
separated.
(ii) Let relint C ∩ relint D =∅. Then C and D are separated.
Proof. (i) By the assumptions in (i), we may choose p ∈ C, q ∈ D having minimum
distance. Let u = q − p (�= o). Then the slab {x : u · p ≤ u · x ≤ u · q} separates C
and D.
(ii) By Proposition 3.1, the sets relint C and relint D are convex. Since, by
assumption, these sets are disjoint, Proposition 4.1 shows that o �∈ E = relint C −
relint D. We shall prove that o �∈ int cl E. Otherwise, there are points x1,...,xd+1 ∈
E such that o is an interior point of the simplex with vertices x1,...,xd+1. Since E is
convex, by Proposition 4.1 we have o ∈ E, which is the desired contradiction. Thus
o ∈ bd cl E or o �∈ cl E. Since E is convex, cl E is convex by Proposition 3.1. It thus
follows from Theorem 4.2, respectively, from (i), that {o} and cl E can be separated
by a hyperplane. Hence, a fortiori, {o} and E = relint C − relint D can be separated
by a hyperplane. Thus relint C and relint D can be separated by a hyperplane by
Proposition 4.1. This, in turn, implies that cl relint C and cl relint D can be separated
by a hyperplane. Now apply Proposition 3.1 to see that C and D can be separated by
a hyperplane. ⊓⊔
Simple examples show that, in general, disjoint closed convex sets C and D
cannot be strongly separated, but if C and D are convex polyhedra, this is possible.
This fact is of importance in optimization (Fig. 4.3).
Strong and Weak Oracles to Specify Convex Bodies
Before considering oracles, some definitions are in order: Let C be a proper convex
body in E d and ε>0. Define the ε-neighbourhood of C or the parallel body of C at
distance ε and the inner parallel body of C at distance ε by
Cε = C + εB d = � x + εy : x ∈ C, y ∈ B d� , C−ε = � x : x + εB d ⊆ C � .