On the Centered Hausdorff Measure of the Sierpinski Gasket

23 Pages Posted: 24 Nov 2021

See all articles by Marta Llorente

Marta Llorente

Universidad Autónoma de Madrid - Department of Economic Analysis: Quantitative Economics

María Eugenia Mera

Universidad Complutense de Madrid (UCM) - Department of Economic Analysis and Quantitative Economics

Manuel Moran

Universidad Complutense de Madrid (UCM) - Department of Economic Analysis and Quantitative Economics

Abstract

We show that the centered Hausdorff measure, Cs(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 Cs(S) is A-computable (algorithmic-computable) in the sense that there exists an algorithm that converges to Cs(S), with error bounds tending to zero. Using this algorithm and bound we show that Cs (S)~1.0049, and we establish a conjecture for the value of the spherical Hausdroff s-measure of S, Hssph(S)~0.8616, and provide an upper bound for it, Hssph(S) ≤ 0.8619.

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

Suggested Citation

Llorente, Marta and Mera, María Eugenia and Moran, Manuel, On the Centered Hausdorff Measure of the Sierpinski Gasket. Available at SSRN: https://ssrn.com/abstract=3970808 or http://dx.doi.org/10.2139/ssrn.3970808

Marta Llorente

Universidad Autónoma de Madrid - Department of Economic Analysis: Quantitative Economics ( email )

Madrid
Spain

María Eugenia Mera

Universidad Complutense de Madrid (UCM) - Department of Economic Analysis and Quantitative Economics ( email )

Madrid
Spain

Manuel Moran (Contact Author)

Universidad Complutense de Madrid (UCM) - Department of Economic Analysis and Quantitative Economics ( email )

Madrid
Spain

HOME PAGE: http://www.ucm.es/info/anaeco/html/manuel.htm

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