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fbtwitterlinkedinvimeoflicker grey 14rssslideshare1
Languages: English
Types: Doctoral thesis
Subjects: QA440
A major theme in geometric measure theory is establishing global properties, such as rectifiability, of sets or measures from local ones, such as densities or tangent measures. In establishing sufficient conditions for rectifiability it is useful to know what local properties are possible in a given setting, and this is the theme of this thesis.\ud \ud It is known, for 1-dimensional subsets of the plane with positive lower density, that the tangent measures being concentrated on a line is sufficient to imply rectifiability. It is shown here that this cannot be relaxed too much by demonstrating the existence of a 1-dimensional subset of the plane with positive lower density whose tangent measures are concentrated on the union of two halflines, and yet the set is unrectiable.\ud \ud A class of metrics are also defined on R, which are functions of the Euclidean metric, to give spaces of dimension s (s > 1), where the lower density is strictly greater than 21-s, and a method for gaining an explicit lower bound for a given dimension is developed. The results are related to the generalised Besicovitch 1/2 conjecture.\ud \ud Set functions are defined that measure how easily the subsets of a set can be covered by balls (of any radius) with centres in the subset. These set functions are studied and used to give lower bounds on the upper density of subsets of a normed space, in particular Euclidean spaces. Further attention is paid to subsets of R, where more explicit bounds are given.
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    • 1 Introduction and Background 6 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Hausdor Measure and Dimension . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.5 Tangent Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
    • 2 Tangent Measures of a One Dimensional Subset of R2 20 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 Approximately Constant Functions . . . . . . . . . . . . . . . . . . . . . . . 21 2.3 A Set from Dickinson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
    • 3 Unrecti able Metric Spaces with High Lower Hausdor s-Densities 43 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2 Construction of the Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3 Density Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.4 Extension to s > 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
    • Theorem 4.5.3 Let E
    • [10] Camillo De Lellis and Felix Otto, Structure of entropy solutions to the eikonal equation, J. Eur. Math. Soc. (JEMS) 5 (2003), no. 2, 107{145. MR 1985613 (2004g:35156)
    • [12] Jozef Dobos, On modi cations of the Euclidean metric on reals, Tatra Mt. Math. Publ. 8 (1996), 51{54, Real functions '94 (Liptovsky Jan, 1994). MR 1475259 (98k:26022)
    • [13] K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986. MR MR867284 (88d:28001)
    • [15] Bernd Kirchheim, Geometry of measures, Ph.D. thesis, University of Prague, 1988.
    • [22] David Preiss and Jaroslav Tiser, On Besicovitch's 12 -problem, J. London Math. Soc. (2) 45 (1992), no. 2, 279{287. MR 1171555 (93d:28012)
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