Schechter, Alexander; (2002) Regularity and other properties of Hausdorff measures. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

In the first part of the thesis the centred Hausdorff measures are studied. These measures are an often used tool in multifractal geometry and are in-creasingly becoming an object of study themselves. The main result of this part is that the centred Hausdorff measures are Borel regular under certain natural conditions. This question has been open for many years and has various interesting applications in the field of multifractal geometry. The significant step in proving this relation is achieved by showing the equivalence of the centred Hausdorff measure and the spherical measure. A counterexample is also given, for showing that this equivalence does not necessarily hold, if certain conditions are not fulfilled. The other part of the thesis concerns the Besicovitch 1/2-problem. This problem has been open since 1928 and it is arguably the most famous questions in classical geometric measure theory. The general version of this conjecture states that a subset of a separable metric space with lower n-density strictly greater than 1/2 almost everywhere is n-rectifiable. There have been partial results since, but the question remained open. A non-rectifiable metric space on the real line is constructed in this thesis, so that the lower 2-density is strictly greater than 1/2. This proves that the generalized Besicovitch conjecture cannot be true. An important tool for the study of the densities is the isodiametric inequality. For various metrics and in particular for the Heisenberg group several results concerning this inequality and the lower densities are proved.

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