Mathematicians David (U. de Paris-Sud-Orsay) and Toro (U. of Washington-Seattle) extend the proof of Reifenberg's topological disk theorem to allow the case of sets with holes, and give sufficient conditions on a set E for the existence of a bi-Lipschitz parametrization of E by a d-dimensional plane or smooth manifold. Their topics include coherent families of balls and planes, Reifenberg-flatness of the image, variants of the Reifenberg theorem, big pieces of bi-Lipschitz images and approximation by bi-Lipschitz domains, and uniform rectifiability and Ahlfore-regular Reifenberg-flat sets. There is no index. Annotation ©2012 Book News, Inc., Portland, OR (booknews.com).