Upper Minkowski dimension estimates for convex restrictions

TL;DR

This paper investigates the Minkowski dimension of subsets where functions in certain Hölder classes are convex, concave, or monotone, revealing limitations on the size of such subsets for various smoothness levels.

Contribution

It constructs functions in Hölder classes with prescribed non-convex or non-monotone restrictions on large-dimensional sets, extending understanding of geometric restrictions of functions.

Findings

Existence of functions with no convex restrictions on sets of Minkowski dimension greater than α-1.

Typical functions in certain Hölder classes have convex restrictions on sets of full Minkowski dimension.

Non-existence of similar properties for monotone restrictions when α<1/2.

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.