On visibility and covering by convex sets (Q1961351)

A set \(X\) is \(n\)-convex if, among any \(n\) of its points, some two of them are connected by a line-segment contained in \(X\). (Thus, a 2-convex set is convex, while a nonconvex (dart-shaped) quadrilateral is 3-convex but not convex.) It is shown that a closed \((n+1)\)-convex set must be the union of at most \(18n^3\) convex sets, improving a 1990 result of Perles and Shelah. A quadratic lower bound on the number of convex sets needed is given. Another standard example of a set that is 3-convex but not convex is obtained by removing a point from the interior of a closed convex set. Results are obtained for \(n\)-convex sets with a specified number of one-point holes, and for sets whose intersection with any vertical line is simply connected. There is also a generous selection of open problems related to \(n\)-convexity.