A POTENTIAL THEORETIC PROOF OF AN INEQUALITY OF C. D. CUTLER AND L. OLSEN

Abstract

We present a potential theoretic proof of the following inequality of C. D. Cutler and L. Olsen [C. D. Cutler & L. Olsen, A Variational Principle for the Hausdorff Dimension of Fractal Sets, Math. Scand.(to appear)]: If $E$ is a Borel subset of ${ℝ}^d$, then

$$\dim E\le \underset{\mu \in 𝒫(E)}{\sup }\underset{¯}{R}(\mu )$$

where dim $E$ denotes the Hausdorff dimension of $E$, $𝒫(E)$ denotes the family of Borel probability measures supported by $E$, and $\underset{\_}{R}(\mu )$ denotes the lower Rényi dimension of the measure $\mu$.