Upper Minkowski dimension estimates for convex restrictions

Abstract

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