Topological methods in combinatorial geometry

This survey is devoted to some results in the area of combinatorial and convex geometry, from classical theorems up to the latest contemporary results, mainly those results whose proofs make essential use of the methods of algebraic topology. Various generalizations of the Borsuk–Ulam theorem for a $(Z_p)^k$-action are explained in detail, along with applications to Knaster's problem about levels of a function on a sphere, and applications are discussed to the Lyusternik–Shnirel'man theory for estimating the number of critical points of a smooth function. An overview is given of the topological methods for estimating the chromatic number of graphs and hypergraphs, in theorems of Tverberg and van Kampen–Flores type. The author's results on the ‘dual’ analogues of the central point theorem and Tverberg's theorem are described. Results are considered on the existence of inscribed and circumscribed polytopes of special form for convex bodies and on the existence of billiard trajectories in a convex body. Results on partition of measures by hyperplanes and other partitions of Euclidean space are presented. For theorems of Helly type a brief overview is given of topological approaches connected with the nerve of a family of convex sets in Euclidean space. Also surveyed are theorems of Helly type for common flat transversals, and results using the topology of the Grassmann manifold and of the canonical vector bundle over it are considered in detail.