Gruber P. Convex and Discrete Geometry

46 Convex Bodies
C ∩ L ⊥ clearly is a closed convex cone with apex o. We show that it is pointed:
(C ∩ L ⊥ ) ∩ � − (C ∩ L ⊥ ) � = (C ∩ L ⊥ ) ∩ (−C) ∩ (−L ⊥ )
= � C ∩ (−C) � ∩ (L ⊥ ∩ L ⊥ ) = L ∩ L ⊥ ={o}. ⊓⊔
Convex Cone Generated by a Set, Positive Hull
If {y1,...,yn} is a finite set in E d , then by the cone generated by it, cone{y1,...,yn},
or by its positive hull pos{y1,...,yn}, we mean the set
� λ1y1 +···+λn yn : λi ≥ 0 � .
It is easy to see that this is a closed convex cone with apex o, in fact, the smallest
such cone containing the set {y1,...,yn}. An open subset C of E d is an open convex
cone with apex o if
λx + µy ∈ C for all x, y ∈ C, λ,µ>0.
Then C is convex. Note that o �∈ C, unless C = E d .
3.2 An Excursion into Combinatorial Geometry: The Theorems
of Carathéodory, Helly and Radon
The following result of Kirchberger [585] of 1903 seems to be the first result of
what is now called combinatorial geometry. It still conveys well the spirit of the
latter. Consider a flock of black and white sheep. If among any four sheep the black
ones can be separated from the white ones by a straight fence, then the black and
the white sheep of the whole flock can thus be separated. The later theorems of
Carathéodory [190], Radon [821] and, in particular, of Helly [490] attracted much
more attention and led to a multitude of pertinent results, especially in the 1960s and
1970s. Some of these results were proved for spaces more general than E d .
We prove below the theorems of Radon, Carathéodory, and Helly. While we have
proved Carathéodory’s theorem in a similar way in Sect. 3.1, the present proof makes
the relation between the theorems of Radon and Carathéodory more clear. These
results, as well as certain other properties of convex sets, led to various attempts
to extend convexity or, rather, combinatorial geometry to a more general context.
The background of Helly’s theorem is singular homology theory, see the interesting
article of Debrunner [249].
For more information we refer to the surveys of Danzer, Grünbaum and Klee
[241] and Eckhoff [283, 284] and to the monograph of Boltyanskiĭ, Martini and
Soltan [145]. From the voluminous literature we cite the article of Tverberg [1003]
on Radon’s theorem. Other aspects of combinatorial geometry were treated by
Goodman, Pollack and Wenger [386] and Matouˇsek [695]. Axiomatic and generalized
convexity is dealt with by Van de Vel [1005] and Coppel [222].