Convexity in combinatorial structures. (English) Zbl 0644.52001

Abstract analysis, Proc. 14th Winter Sch., Srnî/Czech. 1986, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 14, 261-293 (1987).
[For the entire collection see Zbl 0627.00012.]
In this paper the development of abstract convexity since 1968 is surveyed. The attention is restricted to combinatorial structures where the convexity notions are introduced by intrinsic means, i.e. where the definition of convex set only depends on the combinatorial structure itself. Section 2 deals with the major combinatorial themes of convexity theory in real vector spaces. A “prototheory of convexity” is introduced in section 3. Section 4 surveys results in axiomatic convexity, mainly relationships between Carathéodory, Helly and Radon numbers. Convex Geometries, where a convex set is the convex hull of its extreme points, is viewed in section 5 as a part of the theory of Greedoids. Section 6 deals with convex sets on graphs, section 7 with oriented matroids, section 8 with ordered sets, and section 9 with tree- like structures. The paper counts 18 research problems.
Reviewer: G.Sierksma


52-02 Research exposition (monographs, survey articles) pertaining to convex and discrete geometry
52A01 Axiomatic and generalized convexity
52A07 Convex sets in topological vector spaces (aspects of convex geometry)
52A35 Helly-type theorems and geometric transversal theory
05C99 Graph theory
05B35 Combinatorial aspects of matroids and geometric lattices
06A99 Ordered sets


Zbl 0627.00012
Full Text: EuDML