  In this paper we study geometric coincidence problems in the spirit of the
following problems by B. Gr\"unbaum: How many affine diameters of a convex body
in $\mathbb R^n$ must have a common point? How many centers (in some sense) of
hyperplane sections of a convex body in $\mathbb R^n$ must coincide?
  One possible approach to such problems is to find topological reasons for
multiple coincidences for a continuous map between manifolds of equal
dimension. In other words, we need topological estimates for the multiplicity
of a map. In this work examples of such estimates and their geometric
consequences are presented.
