Waist of balls in different spaces

    Gromov and Memarian (2003--2011) have established the waist inequality asserting that for any continuous map f from the sphere    to  there exists a fiber  such that every its t-neighborhood has measure at least the measure of the t-neighborhood of an equatorial subsphere  of .
    Going to the limit we may say that the (n-k)-volume of the fiber  is at least that of the standard sphere . We extend this limit statement to the exact bounds for balls in spaces of constant curvature, tori, parallelepipeds, projective spaces and other metric spaces.
    This talk is based on a joint work with Romans Karasev and Alfredo Hubard.