Lipschitz homotopy convergence of Alexandrov spaces II

We establish a quantitative version of the Lipschitz homotopy convergence introduced by Mitsuishi and Yamaguchi for a moduli space of compact Alexandrov spaces without collapsing. Along the way, we obtain a Lipschitz version of Petersen’s homotopy stability theorem that is applicable to more general settings, including CAT spaces. We also show that the Lipschitz homotopies can be chosen to preserve the singular strata of Alexandrov spaces, i.e., extremal subsets.