Megeney, Alison Claire Verne; (1999) The Besicovitch-Hausdorff dimension of the residual set of packings of convex bodies in Rn. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

I undertake a study of the Besicovitch-Hausdorff dimension of the residual set of arbitrary packings of convex bodies in Rn. In my second chapter, I consider packings of convex bodies of bounded radius of curvature and of fixed orientation into the unit plane square. I show that the Besicovitch-Hausdorff dimension, s, of the residual set of an arbitrary packing satisfies [diagram] where r0 is the bound for the radius of curvature. In chapter 3, I construct a packing which demonstrates that this bound is of the correct order. I generalise the 2-dimensional result to higher dimensions in chapter 4. I use a slicing arguement to prove this. In the final chapter, I tackle the disk packing problem. Using Dirichlet cells, I improve the bound obtained in [1] to 1.033.

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