Upper Minkowski dimension estimates for convex restrictions

Abstract

We show that there are functions $f$ in the Hölder class $C^{\alpha }[0,1]$, $1<\alpha <2$ such that $f{|}_A$ is not convex, nor concave for any $A\subset [0,1]$ with ${\overline{dim}}_{M}A>\alpha -1$. Our earlier result shows that for the typical/generic $f\in C_1^{\alpha }[0,1]$, $0\le \alpha <2$ there is always a set $A\subset [0,1]$ such that $f{|}_A$ is convex and ${\overline{dim}}_{M}A=1$. The analogous statement for monotone restrictions is the following: there are functions $f$ in the Hölder class $C^{\alpha }[0,1]$, $1/2\le \alpha <1$ such that $f{|}_A$ is not monotone on $A\subset [0,1]$ with ${\overline{dim}}_{M}A>\alpha$. This statement is not true for the range of parameters $\alpha <1/2$ and our theorem for the parameter range $1\le \alpha <3/2$ cannot be obtained by integration of the result about monotone restrictions.