Extending Lipschitz functions via random metric partitions

Abstract

Many classical problems in geometry and analysis involve the gluing together of local information to produce a coherent global picture. Inevitably, the difficulty of such a procedure lies at the local boundary, where overlapping views of the same locality must somehow be merged. It is therefore desirable that the boundaries be “smooth,” allowing a graceful transition from one viewpoint to the next. For instance, one may point to Whitney’s use of partitions of unity in studying what is now known as the Whitney extension problem [36, 37]. In the present work, we consider what is perhaps the most basic Whitney-type extension problem, that of extending a Lipschitz function so that it remains Lipschitz. Often such a map is extended by first producing a cover of the new domain, extending the mapping locally, and then gluing together the individual pieces. Our main observation is that in many cases, if one chooses a random cover from the right distribution, the boundary can be made “smooth” on average, even when the local maps are individually quite coarse. This insight leads to the unification, generalization, and improvement of many known results, as well as to new results for many interesting spaces.