Geodesic
A Decomposition for Sets Having a Segment Convexity Property
Canadian journal of mathematics, Tome 32 (1980) no. 1, pp. 21-26

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Let 5 be a subset of Euclidean space. The set 5 is said to be m-convex, m ≥ 2, if and only if for every m distinct points of S, at least one of the line segments determined by these points lies in 5. Clearly any union of m mdash — 1 convex sets will be m-convex, yet the converse is false. However, several decomposition theorems have been proved which allow us to write any closed planar m-convex set as a finite union of convex sets, and actual bounds for the decomposition in terms of m have been obtained ([6], [4], [3]). Moreover, with the restriction that (int cl S) ∼ S contain no isolated points, an arbitrary planar m-convex set S may be decomposed into a finite union of convex sets ([1].Here we strengthen the m-convexity condition to define an analogous combinatorial property for segments. 
Breen, Marilyn. A Decomposition for Sets Having a Segment Convexity Property. Canadian journal of mathematics, Tome 32 (1980) no. 1, pp. 21-26. doi: 10.4153/CJM-1980-002-7
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