Monotone and convex restrictions of continuous functions

TL;DR

This paper investigates the size of subsets of [0,1] where functions in certain Hölder spaces are monotone or convex, revealing generic and universal properties related to the Hausdorff and Minkowski dimensions of such sets.

Contribution

It characterizes the Hausdorff and Minkowski dimensions of large sets where functions are monotone or convex, distinguishing between generic functions and all functions in Hölder spaces.

Findings

For generic functions in $C_{1}^{{eta}}[0,1]$, the dimension of sets where the function is convex or concave is at most $eta - 1$.

For all functions in $C^{{eta}}[0,1]$ with $1< eta extless 2$, there exists a set with Hausdorff dimension $eta - 1$ where the function is convex or concave.

Results highlight differences between typical and universal behaviors of functions regarding monotonicity and convexity on large sets.

Abstract

Suppose that $f$ belongs to a suitably defined complete metric space $ {{\cal C}}^{{\alpha}}$ of H\"older $ {\alpha}$-functions defined on $[0,1]$. We are interested in whether one can find large (in the sense of Hausdorff, or lower/upper Minkowski dimension) sets $A {\subset} [0,1]$ such that $f|_{A}$ is monotone, or convex/concave. Some of our results are about generic functions in $ {{\cal C}}^{{\alpha}}$ like the following one: we prove that for the generic $f\in C_{1}^{{\alpha}}[0,1]$, $0\leq {\alpha}<2$ for any $A {\subset} [0,1]$ such that $f|_{A}$ is convex, or concave we have ${\mathrm{dim}}_{\mathrm H} A\leq \underline{\mathrm{dim}}_M A\leq \max \{0, {\alpha}-1 \}.$ On the other hand, we also have some results about all functions belonging to a certain space. For example the previous result is complemented by the following one: for $1< {\alpha}\leq 2$ for any $f\in C^{{\alpha}}[0,1]$ there is always a set $A {\subset}[0,1]$ such that ${\mathrm{dim}}_{\mathrm H} A= {\alpha}-1$ and $f|_{A}$ is convex, or concave on $A$.