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 1.2 and Proposition 1.4).
Extremal Point Convex Subset Chromatic Number Relative Interior Vertical Segment
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