Multifractal properties of convex hulls of typical continuous functions

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 ${\mathbf{E}}_{\phi }^h$ the set of points at which $\phi :[0,1{]}^d\to \mathbb{R}$ has a pointwise Hölder 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 ${\phi }_{1,f}(\mathbf{x})=\max \{y:(\mathbf{x},y)\in H_f\}$ and ${\phi }_{2,f}(\mathbf{x})=\min \{y:(\mathbf{x},y)\in H_f\}$ denotes the convex function on the bottom of $H_f$. We show that there is a dense $G_{\delta }$ subset $\mathcal{G}\subset C[0,1{]}^d$ such that for $f\in \mathcal{G}$ the following properties are satisfied. For $i=1,2$ the functions ${\phi }_{i,f}$ and $f$ coincide only on a set of zero Hausdorff dimension, the functions ${\phi }_{i,f}$ are continuously differentiable on $(0,1{)}^d$, ${\mathbf{E}}_{{\phi }_{i,f}}^0$ equals the boundary of $[0,1{]}^d$, ${\dim }_H{\mathbf{E}}_{{\phi }_{i,f}}^1=d-1$, ${\dim }_H{\mathbf{E}}_{{\phi }_{i,f}}^{+\infty }=d$ and ${\mathbf{E}}_{{\phi }_{i,f}}^h=\varnothing$ if $h\in (0,+\infty )\setminus \{1\}$.