A refinement of theorems of Kirchberger and Carathéodory

Abstract

I can indicate the type of refinement mentioned in the title by referring to Kirchberger's theorem [4]. Its picturesque form in the plane is: if sheep and goats are grazing in a field and for every four animals there exists a line separating the sheep from the goats then there exists such a line for all the animals. The refinement is that the words ‘every four animals’ may be replaced by ‘every four animals including an arbitraily chosen animal’; this reduces the ‘Kirchberger number’ from four to, effectively, three.

Mapping toric varieties into low dimensional spaces

“…As J + is nonempty and disjoint from J − , there exist j 0 ∈ J + \J − . By Watson's Carathéodory Theorem [28], there exists a subset K ⊆ I ′ with |K| r and K ∩ J − = ∅ such that m ′ is in the convex hull of {m j0 } ∪ {m k | k ∈ K}, that is, there are nonnegative rational numbers µ j0 , µ k such that k∈K∪{j0} µ k = 1 and…”

Covering a polygon with triangles: a Caratheodory-type theorem

“…This paper pursues a train of thought suggested by the theorems of Caratheodory (1907) and Watson (1973). In two dimensions we ask how thickly covered with triangles is the convex hull of a family of points, the vertices of the triangles being points of the family.…”

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AbstractGeneralizations are proved for theorems of Caratheodory (1907), Kirchberger (1903) and Watson (1973), the theme of these results being how thickly the convex hull of a family of points is covered by simplexes whose vertices are chosen from the points of the family.

Summary

This paper pursues a train of thought suggested by the theorems of Caratheodory (1907) andWatson (1973).

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