Rejoinder-A Gibbs Sampler for a Class of Random Convex Polytopes

We are very grateful to all commenters for their stimulating remarks, questions, as well as useful pointers to the literature which span a wide range of statistical methods over decades of research. We have neither the space nor the knowledge to answer many of the questions raised, and we only aim to offer some clarifications. We hope that readers will be as enthusiastic as ourselves about research on the topics discussed by the commenters. In the following, we refer to Diaconis and Wang as DW, Hoffman, Hannig and Zhang as HHZ, Lawrence and Vander Wiel as LV, Ruggeri as R, Shafer as S, and Williams as W.