Convexity in combinatorial structures. (English) Zbl 0644.52001

Duchet, Pierre

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

MSC:

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

Keywords:

exchange number; Carathéodory number; Helly number; Radon number; Eckhoff’s conjecture; Sierksma number; axiomatic convexity; convex sets on graphs; oriented matroids; ordered sets; tree-like structures; research problems

Citations:

Zbl 0627.00012