Gruber P. Convex and Discrete Geometry

254 Convex Polytopes
The Face Lattices of P and P ∗ are Anti-Isomorphic
We first present a tool, where
is the polar of P, see Sect. 9.1.
P ∗ = � y : x · y ≤ 1forx ∈ P �
Proposition 14.2. Let P ∈ P such that o ∈ int P. Then the following statements
hold:
(i) P ∗ ∈ P.
(ii) P ∗∗ = P.
Proof. (i) Represent P in the form P = conv{x1,...,xn}. Then
P ∗ = � y : (λ1x1 +···+λnxn) · y ≤ 1forλi ≥ 0, λ1 +···+λn = 1 �
= � y : xi · y ≤ 1fori = 1,...,n} =
n�
{y : xi · y ≤ 1 �
is an intersection of finitely many closed halfspaces. Since o ∈ int P, there is ϱ>0
such that ϱB d ⊆ P. The definition of polarity then easily yields P ∗ ⊆ (ϱB d ) ∗ =
(1/ϱ)B d . Thus P ∗ is bounded. Since P ∗ is the intersection of finitely many halfspaces
and is bounded, Theorem 14.2 implies that P ∗ ∈ P.
(ii) To show that P ⊆ P ∗∗ ,letx ∈ P. Then x · y ≤ 1 for all y ∈ P ∗ by the
definition of P ∗ . This, in turn, implies that x ∈ P ∗∗ by the definition of P ∗∗ .To
show that P ⊇ P ∗∗ ,letx ∈ E d \P. The separation theorem 4.4 then provides a point
y ∈ E d such that x · y > 1 while z · y ≤ 1 for all z ∈ P. Hence y ∈ P ∗ .Fromx · y > 1
we then conclude that x �∈ P ∗∗ by the definition of P ∗∗ . The proof that P = P ∗∗ is
complete. ⊓⊔
The following result relates the face lattices of P and P ∗ , where dim ∅=−1.
Theorem 14.7. Let P ∈ P such that o ∈ int P. Define a mapping ♦ = ♦P by
i=1
F ♦ = � y ∈ P ∗ : x · y = 1 for x ∈ F � for F ∈ F(P),
where, in particular, ∅ ♦ = P ∗ and P ♦ =∅. Then the following statements hold:
(i) ♦ is a one-to-one mapping of F(P) onto F(P ∗ ).
(ii) dim F ♦ + dim F = d − 1 for each F ∈ F(P).
(iii) ♦ is an anti-isomorphism of the lattice � F(P), ∧, ∨ � onto the lattice
� F(P ∗ ), ∧, ∨ � .
By the latter we mean that ♦ is one-to-one and onto and
(F ∧ G) ♦ = F ♦ ∨ G ♦ and (F ∨ G) ♦ = F ♦ ∧ G ♦ for F, G ∈ F(P).