On the Centred Hausdorff Measure of the Sierpinski Gasket

Abstract

We show that the centred Hausdorff measure, $C^{s}(S),$ with $s=\frac{\log3}{\log2},$ of the Sierpinski gasket $S$, is $C$-computable (continuous-computable), in the sense that its value is the solution of the minimisation problem of a continuous function on a compact domain. We also show that $C^{s}(S)$ is $A$-computable (algorithmic-computable) in the sense that there is an algorithm that converges to $C^{s}(S),$ with error bounds tending to zero. Using this algorithm and bounds we show that $C^{s}(S)\sim1.0049,$ and we establish a conjecture for the value of the spherical Hausdorff $s$-measure of $S,$ $\mathcal{H}_{sph}^{s}(S)\sim0.8616$, and provide an upper bound for it, $\mathcal{H}_{sph}^{s}(S)\leq0.8619.$