Multifractal properties of convex hulls of typical continuous functions

TL;DR

This paper investigates the multifractal spectrum of convex hulls of typical continuous functions on [0,1]^d, revealing detailed geometric and differentiability properties of associated functions.

Contribution

It characterizes the multifractal spectrum and differentiability of convex hulls of generic continuous functions, providing new insights into their geometric structure.

Findings

Functions coincide with their convex/concave hulls only on zero Hausdorff dimension sets.

Convex/concave hull functions are continuously differentiable on (0,1)^d.

Multifractal spectrum is explicitly determined, with Hausdorff dimensions of level sets.

Abstract

We study the singularity (multifractal) spectrum of the convex hull of the typical/generic continuous functions defined on $[0,1]^{d}$. We denote by ${\mathbf E}_ { { \varphi } }^{h} $ the set of points at which $ \varphi : [0,1]^d\to {\mathbb R}$ has a pointwise H\"older exponent equal to $h$. Let $H_{f}$ be the convex hull of the graph of $f$, the concave function on the top of $H_{f}$ is denoted by $ { { \varphi } }_{1,f}( { { \mathbf x } })=\max \{y:( { { \mathbf x } },y)\in H_{f} \}$ and $ { { \varphi } }_{2,f}( { { \mathbf x } })=\min \{y:( { { \mathbf x } },y)\in H_{f} \}$ denotes the convex function on the bottom of $H_{f}$. We show that there is a dense $G_\delta$ subset $ { { \cal G } } { \subset } {C[0,1]^d}$ such that for $f\in { { \cal G } }$ the following properties are satisfied. For $i=1,2$ the functions $ { { { \varphi } }_ {i,f}}$ and $f$ coincide only on a set of zero Hausdorff dimension, the functions $ { { { \varphi } }_ {i,f}}$ are continuously differentiable on $(0,1)^{d}$, ${\mathbf E}_{ { { \varphi } }_{i,f}}^{0} $ equals the boundary of $ {[0,1]^d}$, $\dim_{H}{\mathbf E}_{ { { \varphi } }_{i,f}}^{1}=d-1 $, $\dim_{H}{\mathbf E}_{ { { \varphi } }_{i,f}}^{+ { \infty }}=d $ and ${\mathbf E}_{ { { \varphi } }_{i,f}}^{h}= { \emptyset }$ if $h\in(0,+ { \infty }) { \setminus } \{1 \}$.