(based on the joint work with Roman~Karasev and Alfredo~Hubard)
Gromov and Memarian (2003--2011) have established the waist inequality asserting that for any continuous map f:S^n\to R^{n-k} there exists a fiber f^{-1}(y) such that every its t-neighborhood has measure at least the measure of the t-neighborhood of an equatorial subsphere S^k\subset S^n.
Going to the limit we may say that the (n-k)-volume of the fiber f^{-1}(y) is at least that of the standard sphere S^{k}. We extend this limit statement to the exact bounds for balls in spaces of constant curvature, tori, parallelepipeds, projective spaces and other metric spaces.
By the volume of preimages for a non-regular map f we mean its lower Minkowski content, some new properties of which will be also presented in the talk.
