An isodiametric problem of fractal dimension

Abstract

For $0<\lambda <\frac{1}{2}$ we consider the product $F_\lambda =E_\lambda \times {\mathbb R}$, where $E_\lambda$ is the attractor of the IFS $\{x\mapsto \lambda x,\ \ x\mapsto 1\!-\!\lambda +\lambda x\}$ on ${\mathbb R}$. The Huasdorff dimension of $F_\lambda$ is $s=1-\frac{\ln 2}{\ln \lambda }$. We show that $\sup \left\{ \frac{{\mathcal H}^s(X\cap F_\lambda )}{|X|^s}: |X|>0\right\} =1$ and that there is a convex compact set $A$ ($=A(\lambda )$) with $\frac{{\mathcal H}^s(A\cap F_\lambda )}{|A|^s}=1$. Such a convex compact set $A$ is called an “extremal set” of $F_\lambda$ with respect to $s$-dimensional Hausdorff measure ${\mathcal H}^s$. When $\lambda$ is small, say $\lambda \le \frac{1}{5}$, we further show that there exists an extremal set $A$ with $|A|\ge \frac{2}{\sqrt{3}}$ such that ${\mathcal H}^s(A\cap F_\lambda )={\mathcal H}^s(D_{|A|}\cap F_\lambda )$ for $D_{|A|}=\left\{ (x,y): \left( x-\frac{1}{2}\right) ^2+y^2 \le \frac{1}{4}|A|^2\right\}$. As an application, we can estimate the value of ${\mathcal H}^s(E_\lambda \!\times \![0,1])$ to any pre-set error $\epsilon$.