Gruber P. Convex and Discrete Geometry

50 Convex Bodies
The Centrepoint of a Finite Set
We give a second application of Helly’s theorem. Let A ⊆ Ed be a finite set consisting
of, say, n points. A point x ∈ Ed is a centrepoint of A if each closed halfspace
n
which contains x contains at least d+1 points of A. Rado [820] proved the following
result.
Theorem 3.4. Each finite set in E d has a centrepoint.
Proof. Let A ⊆ Ed be a finite set with n points. Clearly, a point in Ed is a centrepoint
of A if and only if it lies in each open halfspace H o which contains more than d
d+1 n
points of A. For the proof of the theorem it is thus sufficient to show that the open
halfspaces H o have non-empty intersection. This is certainly the case if the convex
polytopes conv(A ∩ H o ) have non-empty intersection. Given d + 1 of these polytopes,
each contains more than d
d+1 n points of A. If their intersection were empty,
then each point of A is in the complement of one of these polytopes. Hence A is
the union of these complements, but the union consists of less than n points. This
contradiction shows that any d + 1 of the polytopes have non-empty intersection. By
Helly’s theorem all these polytopes then have non-empty intersection. Each point of
this intersection is a centrepoint. ⊓⊔
Remark. For extensions and the related ham-sandwich theorem see Matouˇsek [695]
and the references cited there.
3.3 Hartogs’ Theorem on Power Series
Given a power series in d complex variables,
(1)
∞�
an1···nd zn1
1 ···znd d for z = (z1,...,zd) ∈ C d ,
n1,...,nd=0
the problem arises to determine the largest connected open set in C d on which it
converges, its domain of convergence. Considering the case d = 1, it is plausible to
conjecture, that a domain of convergence is a polycylinder, i.e. a Cartesian product
of d open circular discs. Surprisingly, this is wrong. A theorem of Hartogs gives a
complete description of domains of convergence.
This section contains the definition of Reinhardt domains and Hartogs’ theorem
which says that the domains of convergence of power series in d complex variables
are precisely the Reinhardt domains. Only part of the proof is given.
For a complete proof and for more information on analytic functions in several
complex variables, see, e.g. L. Kaup and B. Kaup [568].
Reinhardt Domains and Domains of Convergence
In order to state Hartogs’ theorem, we need the notion of a (complete, logarithmically
convex) Reinhardt domain. This is an open, connected set G in C d with the following
properties: