Carathéodory and Helly-numbers of convex-product-structures

Gerard Sierksma

Vol. 61 (1975), No. 1, 275–282

DOI: 10.2140/pjm.1975.61.275

Abstract

Let c₁ and c₂ be the Carathéodory-numbers of the convexity-structures 𝒞₁ for X₁, respectivily 𝒞₂ for X₂. It is shown that the Carathéodory-number c of the convex-product-structure 𝒞₁ ⊕𝒞₂ for X₁ × X₂ satisfies the inequality c₁ + c₂ − 2 ≦ c ≦ c₁ + c₂; c₁,c₂ ≧ 2.

The upper bound for c can be improved by one, resp. two, if a certain number, namely the so-called exchange-number, of one resp. each of the structures 𝒞₁ and 𝒞₂ is less than or equal to the Carathéodory-number of that structure.

A new definition of the Helly-number is given and Levi’s theorem is proved with this new definition. Finally it is shown that the Helly-number of a convex-product-structure is the greater of the Helly-numbers of 𝒞₁ and 𝒞₂.

Mathematical Subject Classification 2000

Primary: 52A40

Milestones

Received: 2 January 1975

Published: 1 November 1975