Graph explorer

Three-monotone interpolation

A function $f\colon\mathbb R\to\mathbb R$ is called \emph{$k$-monotone} if it is $(k-2)$-times differentiable and its $(k-2)$nd derivative is convex. A point set $P\subset\mathbb R^2$ is \emph{$k$-monotone interpolable} if it lies on a graph of a $k$-monotone function. These notions have been studied in analysis, approximation theory etc. since the 1940s. We show that 3-monotone interpolability is very non-local: we exhibit an arbitrarily large finite $P$ for which every proper subset is $3$-monotone interpolable but $P$ itself is not. On the other hand, we prove a Ramsey-type result: for every $n$ there exists $N$ such that every $N$-point $P$ with distinct $x$-coordinates contains an $n$-point $Q$ such that $Q$ or its vertical mirror reflection are $3$-monotone interpolable. The analogs for $k$-monotone interpolability with $k=1$ and $k=2$ are classical theorems of Erdős and Szekeres, while the cases with $k\ge4$ remain open. We also investigate the computational complexity of deciding $3$-monotone interpolability of a given point set. Using a known characterization, this decision problem can be stated as an instance of polynomial optimization and reformulated as a semidefinite p