Gruber P. Convex and Discrete Geometry

|z2|
3 Convex Sets, Convex Bodies and Convex Hulls 51
log |z2|
|z1| log |z1|
Fig. 3.1. Logarithmically convex sets in C 2
(2) z = (z1,...,zd) ∈ C d ,(w1,...,wd) ∈ G,
|z1| < |w1|,...,|zd| < |wd| ⇒z ∈ G,
(3) �� log |z1|,...,log |zd| � : (z1,...,zd) ∈ G, zi �= 0 � ⊆ E d is convex.
For d = 1, Reinhardt domains are simply the open circular discs with centre
0inC (Fig. 3.1).
Hartogs’ Theorem
Hartogs [480] found the following characterization of domains of convergence of
power series in d complex variables.
Theorem 3.5. Let G be an open connected set in C d . Then the following statements
are equivalent:
(i) G is the domain of convergence of a power series in d complex variables of the
form (1).
(ii) G is a complete, logarithmically convex Reinhardt domain.
We prove only the implication (i)⇒(ii). By const, a positive constant is meant. If
const appears several times in the same context, this does not mean that it is always
the same constant.
Proof. (i)⇒(ii) The main tool for the proof is the following lemma of Abel, see,
e.g. [568]:
(4) Let z,w ∈ Cd be such that |an1···nd wn1
1 ···wnd
for n1,...,nd = 0, 1,..., and let |zi| < |wi| for i = 1,...,d. Then
z ∈ G.
d |≤const
Assume now that G satisfies (i). G is open and connected and, by (4), has property
(2). In order to show that it also has property (3), it is sufficient to show the
following.
(5) Let x, y ∈ G, xi, yi �= 0fori = 1,...,d, let0≤ λ ≤ 1, and let z ∈ C d
such that log |zi| =(1 − λ) log |xi|+λ log |yi| for i = 1,...,d.
Then z ∈ G.