Vol. 61, No. 1, 1975

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Carathéodory and Helly-numbers of convex-product-structures

Gerard Sierksma

Vol. 61 (1975), No. 1, 275–282

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
Received: 2 January 1975
Published: 1 November 1975
Gerard Sierksma