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}_ { { \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 $h\in(0,+ { \infty }) { \setminus } \{1 \}$.