Gruber P. Convex and Discrete Geometry

34 Koebe’s Representation Theorem for Planar Graphs 507
ϑv = ϑv(ρ) = �
vw∈E(G∧ arctan
)
ρw
− π for v ∈ V(G
ρv
∧ ), v �= v1,...,vk, ∞,
�
= ϑvi (ρ) = arctan ρw
− αi
for i = 1,...,k.
2
ϑvi
The quantity
v i w∈E(G ∧ )
w�=∞
ρvi
µ(ρ) = �
v∈V(G ′ )
ϑv(ρ) 2
is a measure for the deviation of the list ρ from a list as postulated in (9). For the
proof of (9), we have to find a list ρ of positive numbers such that µ(ρ) = 0.
First, the following equality will be shown:
(10) �
ϑv(ρ) = 0 for any list ρ = (ρv) of positive numbers.
v∈V(G ′ )
To see (10), note that
(11) �
ϑv(ρ) = �
v∈V(G ′ )
vw∈E(G ′ )
arctan ρw
ρv
− π � #V(G ′ ) − k � − 1
2
+ arctan ρv
ρw
k�
αi.
Since all countries of G ∧ are quadrangles, it follows from Corollary 15.1 that
(12) 2#V(G ′ ) = #E(G ∧ ) + 2 = #E(G ′ ) + k + 2.
Noting that arctan t +arctan (1/t) = π/2fort > 0, Propositions (11) and (12) imply
the equality (10).
Let
∅ �= S � V(G ′ ), l = #{v1,...,vk}∩S.
Clearly, # � S ∪{∞} � = #(S) + 1 and #E � G ∧ (S ∪{∞}) � = #E � G ∧ (S) � + l. An application
of (6) to S ∪{∞}instead of S (for #S ≥ 4 and l ≥ 0) and simple arguments
(for #S = 2, 3 and l = 0) show that
(13) 2#S − #E � G ∧ (S) � ≥ l + 3for#S ≥ 4, l ≥ 0or#S = 2, 3, l = 0.
Simple arguments also yield the following:
i=1
(14) 2#S − #E � G ∧ (S) � ≥ l + 2for#S = 2, 3, l > 0.
Second, we show that the quantity µ(·) attains its minimum. Let R be the set
of all lists ρ = � ρv : v ∈ V(G ′ ) � , where 0 < ρv ≤ 1 and such that ρv = 1
if ϑv(ρ) > 0. In addition, we require that ρv = 1 for at least one v. Wehave
R �= ∅, since the list (1) ∈ R. Unfortunately, R is not compact. Let � ρ (n)� be a
sequence of lists in R such that µ � ρ (n)� tends to the infimum of µ(ρ) for ρ ∈ R.
Apply the Bolzano–Weierstrass theorem to see that, by considering a suitable subsequence
and renumbering, if necessary, the sequence � ρ (n) �
v converges for each
v ∈ V(G ′ ). Let