This paper by Mikhael Gromov discusses filling inequalities in Riemannian manifolds, focusing on isosystolic inequalities, filling radii, and filling volumes. It begins by defining essential manifolds and presents the main isosystolic inequality, which provides a bound on the length of the shortest noncontractible closed curve in terms of the volume of the manifold. The paper then explores the concept of filling radius, which measures how close a manifold can be approximated by a submanifold in a larger space. It also introduces filling volume, which is the minimal volume required to fill a manifold with a higher-dimensional chain. The paper discusses various results and theorems related to these concepts, including the isoperimetric inequality, which relates the volume of a manifold to the volume of its boundary. It also examines the relationship between filling radius and filling volume, showing how they are connected through geometric and topological properties of manifolds. The paper covers the method of Federer-Fleming, which is used to derive isoperimetric inequalities for submanifolds in Euclidean space. It also discusses the filling inequalities in Banach spaces and the relationship between filling radii and injectivity radii. The paper concludes with various examples and applications, including the study of surfaces, minimal geometric cycles, and the behavior of geodesics in manifolds. Overall, the paper provides a comprehensive overview of filling inequalities in Riemannian manifolds, highlighting the interplay between geometric and topological properties and the significance of these inequalities in understanding the structure of manifolds.This paper by Mikhael Gromov discusses filling inequalities in Riemannian manifolds, focusing on isosystolic inequalities, filling radii, and filling volumes. It begins by defining essential manifolds and presents the main isosystolic inequality, which provides a bound on the length of the shortest noncontractible closed curve in terms of the volume of the manifold. The paper then explores the concept of filling radius, which measures how close a manifold can be approximated by a submanifold in a larger space. It also introduces filling volume, which is the minimal volume required to fill a manifold with a higher-dimensional chain. The paper discusses various results and theorems related to these concepts, including the isoperimetric inequality, which relates the volume of a manifold to the volume of its boundary. It also examines the relationship between filling radius and filling volume, showing how they are connected through geometric and topological properties of manifolds. The paper covers the method of Federer-Fleming, which is used to derive isoperimetric inequalities for submanifolds in Euclidean space. It also discusses the filling inequalities in Banach spaces and the relationship between filling radii and injectivity radii. The paper concludes with various examples and applications, including the study of surfaces, minimal geometric cycles, and the behavior of geodesics in manifolds. Overall, the paper provides a comprehensive overview of filling inequalities in Riemannian manifolds, highlighting the interplay between geometric and topological properties and the significance of these inequalities in understanding the structure of manifolds.