Properties of the intrinsic flat distance

In this paper written in honor of Yuri Burago, we explore a variety of properties of intrinsic flat convergence. We introduce the sliced filling volume and interval sliced filling volume and explore the relationship between these notions, the tetrahedral property and the disappearance of points under intrinsic flat convergence. We prove two new Gromov–Hausdorff and intrinsic flat compactness theorems including the Tetrahedral Compactness Theorem. Much of the work in this paper builds upon Ambrosio–Kirchheim's Slicing Theorem combined with an adapted version Gromov's Filling Volume. We are grateful to have been invited to submit a paper in honor of Yuri Burago, in thanks not only for his beautiful book written jointly with Dimitri Burago and Sergei Ivanov but also for his many thoughtful communications with us and other young mathematicians over the years.