Gromov and Memarian (2003--2011) have established the waist inequality asserting that for any continuous map f from the sphere 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 of S^n.Going to the limit we may say that the (n-k)-volume of the fiberf^-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.(based on the joint work with Roman Karasev and Alfredo Hubard)