  We propose the Hit-and-Run algorithm for planning and sampling problems in
non-convex spaces. For sampling, we show the first analysis of the Hit-and-Run
algorithm in non-convex spaces and show that it mixes fast as long as certain
smoothness conditions are satisfied. In particular, our analysis reveals an
intriguing connection between fast mixing and the existence of smooth
measure-preserving mappings from a convex space to the non-convex space. For
planning, we show advantages of Hit-and-Run compared to state-of-the-art
planning methods such as Rapidly-Exploring Random Trees.
