Multifractal properties of typical convex functions

Abstract

We study the singularity (multifractal) spectrum of continuous convex functions defined on $[0,1{]}^d$. Let $E_f(h)$ be the set of points at which $f$ has a pointwise exponent equal to $h$. We first obtain general upper bounds for the Hausdorff dimension of these sets $E_f(h)$, for all convex functions $f$ and all $h\ge 0$. We prove that for typical/generic (in the sense of Baire) continuous convex functions $f:[0,1{]}^d\to \mathbb{R}$, one has $\dim E_f(h)=d-2+h$ for all $h\in [1,2],$ and in addition, we obtain that the set $E_f(h)$ is empty if $h\in (0,1)\cup (1,+\infty )$. Also, when $f$ is typical, the boundary of $[0,1{]}^d$ belongs to $E_f(0)$.