## Quasi-metric geometry: smoothness and convergence results

##### Abstract

This thesis has two distinct yet related parts, the first pertaining to geometry on quasi-metric spaces with emphasis on the Hausdorff outer-measure, the natural extension of the Gromov-Pompeiu-Hausdorff distance to quasi-metric spaces, and smoothness of quasi-metric spaces, and the second dealing with Euclidean geometry, namely the role geometry plays in analysis, particularly in the characterization of Lipschitz domains via cones, domains of class C¹, [superscript w], where w is a modulus of continuity, via pseudo-balls, which includes Lyapunov domains, a sharp version of the Hopf-Oleinik boundary point principle, and subsequently the strong maximum principle.

##### Degree

M.A.