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 $$f:[0,1]^{d}\rightarrow { \mathbb {R}}$$ , 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 $$E_{f}({0})$$ .