Multifractal properties of typical convex functions

TL;DR

This paper investigates the multifractal spectrum of typical convex functions on [0,1]^d, establishing bounds and exact dimensions of singularity sets, revealing a precise multifractal structure for generic convex functions.

Contribution

It provides the first detailed multifractal analysis of typical convex functions, including exact Hausdorff dimension results for their singularity sets.

Findings

For typical convex functions, the dimension of the set where the pointwise exponent is h equals d-2+h for h in [1,2].

The sets of points with exponents outside [1,2] are empty for typical convex functions.

The boundary of the domain belongs to the set of points with zero pointwise exponent for 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 $h\geq 0$. 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 $E_{f}({0})$.