On the Centered Hausdorff Measure of the Sierpinski Gasket

Abstract

We show that the centered Hausdorff measure, C^s(S), with s=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 minimization 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 exists an algorithm that converges to C^s(S), with error bounds tending to zero. Using this algorithm and bound we show that C^s (S)~1.0049, and we establish a conjecture for the value of the spherical Hausdroff s-measure of S, H^s_sph(S)~0.8616, and provide an upper bound for it, H^s_sph(S) ≤ 0.8619.

Keywords: Self similar sets, Sierpinski gasket, Hausdorff measures, density of measures, computability of fractal measures