Axiomatic convexity theory and relationships between the Carathéodory, Helly, and Radon numbers

Partition numbers for trees and ordered sets

“…The above result is a generalization of the Kay and Womble inequality [10] and was also presented by the author at the 1976 Oberwolfach Conference.…”

Generalised Helly and Radon numbers

“…PROOF: To show (a) we first notice the following simple consequence of the Lemma: The proof of (b) is also straightforward. D [3] Helly and Radon numbers 431…”

“…430 K. Kolodziejczyk [2] The (generalised) Helly and Radon numbers play a very important part in the theory of axiomatic convexity and combinatorial geometry. The numbers, their properties and relationships have been investigated in many papers; see, among others, [1,2,3,4,5,9,10,11,12].…”

Carathéodory and Helly-numbers of convex-product-structures

“…Existing notations and definitions have been taken from [3], [4] and, in particular, from [8]. Let ^ be a collection of subsets of a set 1; by Π^7 and U ^ we denote the intersection and the union respectivily, of the elements of ^.…”

“…Of course, if the exchange-number e of the convexity-structure & for X exists then e ^ 1; if ^ is a TΊ-convexity-structure (see [4]) then e^2; if id, |A|^e and pe 9f(A) then %? (A) = \J{^(PV(A\a))\ae A}, see [5], axiom C3; if the Caratheodory-number c of ^ exists too, then e <^ c + 1, which follows directly from Lemma 2.1(ii).…”