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Longest k-monotone chains
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abstract
We study higher order convexity properties of random point sets in the unit square. Given $n$ uniform i.i.d random points, we derive asymptotic estimates for the maximal number of them which are in $k$-monotone position, subject to mild boundary conditions. Besides determining the order of magnitude of the expectation, we also prove strong concentration estimates. We provide a general framework that includes the previously studied cases of $k=1$ (longest increasing sequences) and $k=2$ (longest convex chains).
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Longest convex chains with i.i.d. points
For i.i.d. points with a bounded continuous density in a triangle, the longest convex chain grows like c n^{1/3}, with c given by a density-weighted affine arclength variational formula.
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