A setS inRdis said to bem-convex,m≧2, if and only if for everym distinct points inS, at least one of the line segments determined by these points lies inS. Clearly any union ofm−1 convex sets ism-convex, yet the converse is false and has inspired some interesting mathematical questions: Under what conditions will anm-convex set be decomposable intom−1 convex sets? And for everym≧2, does there exist aσ(m) such that everym-convex set is a union ofσ(m) convex sets? Pathological examples convince the reader to restrict his attention to closed sets of dimension≦3, and this paper provides answers to the questions above for closed subsets of the plane.

General decomposition theorems form-convex sets in the plane

We address the structure of nonconvex closed subsets of the Euclidean plane. A closed subsetS⊆ℝ2 which is not presentable as a countable union of convex sets satisfies the following dichotomy:
 We also show that iff: [N]2→{0, 1} is a continuous coloring of pairs, and no open subset ofN isf-monochromatic, then the least numberκ off-monochromatic sets required to coverN satisfiesK
 +>-c. Consequently, a closed subset of ℝ2 that cannot be covered by countably many convex subsets, cannot be covered by any number of convex subsets other than the continuum or the immediate predecessor of the continuum. The analogous fact is false for closed subsets of ℝ3.

Convex decompositions in the plane and continuous pair colorings of the irrationals

It is shown that if a closed setS in the plane is (n+1)-convex, then it has no more thann 4 holes. As a consequence, it can be covered by≤n 6 convex subsets. This is an improvement on the known bound of 2 n ·n 3.

A closed (n + 1)-convex set inR 2 is a union ofn 6 convex sets

A subset of a vector space is called countably convex if it is a countable union of convex sets. Classification of countably convex subsets of topological vector spaces is addressed in this paper. An ordinal-valued rank function % is introduced to measure the complexity of local nonconvexity points in subsets of topological vector spaces. Then % is used to give a necessary and sufficient condition for countable convexity of closed sets. Theorem. Suppose that S is a closed subset of a Polish linear space. Then S is countably convex if and only if there exists α 2 and are seen to be drastically more complicated than uncountably convex closed subsets of R2.

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Convexity ranks in higher dimensions

A setX⊆ℝd isn-convex if among anyn of its points there exist two such that the segment connecting them is contained inX Perles and Shelah have shown that any closed (n+1)-convex set in the plane is the union of at mostn6 convex sets We improve their bound to 18n3, and show a lower bound of order Ω(n2) We also show that ifX⊆ℝ2 is ann-convex set such that its complement has λ one-point path-connectivity components, λ<∞, thenX is the union ofO(n4+n2λ) convex sets Two other results onn-convex sets are stated in the introduction (Corollary 12 and Proposition 14)

On visibility and covering by convex sets

All the sets with which we are concerned are subsets of the real Euclidean plane E2. By Lm we denote those subsets X of E2 for which, if pl, p2, …, Pm are any m points of X, then at least one segment pipj, i ≠ j consists entirely of points of X. L2 is the class of convex subsets of E2. We shall show that if X is closed and X ∈ Lm. then X is the union of finitely many convex sets. This extends a result of Valentine (4). See also (1),(2),(3).

H. G. Eggleston

Papers

A condition for a compact plane set to be a union of finitely many convex sets