On the Centred Hausdorff Measure of the Sierpinski Gasket

Abstract

We show that the centred Hausdorff measure, $C^s(S),$ with $s=\frac{\log 3}{\log 2},$ 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 $C^s(S)\sim 1.0049,$ and we establish a conjecture for the value of the spherical Hausdorff $s$-measure of $S,$ ${\mathcal{H}}_{sph}^s(S)\sim 0.8616$, and provide an upper bound for it, ${\mathcal{H}}_{sph}^s(S)\le \mathrm{0.8619.}$

31 pages, 7 figures, 1 table