Multifractal properties of convex hulls of typical continuous functions

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 $ \varphi : [0,1]^d\to {\mathbb R}$hasapointwiseH\ddot{o}lderexponentequalto$h$\mathrm{.}Let$H_{f}$betheconvexhullofthegraphof$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}$suchthatfor$f\in { { \cal 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 $ { { { \varphi } }_ {i,f}}$arecontinuouslydifferentiableon$(0,1)^{d}$,${\mathbf E}{ { { \varphi } }{i,f}}^{0} $equalstheboundaryof$ {[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 }$.