A lower bound on the waist of unit spheres of uniformly convex normed spaces

In this paper we give a lower bound on the waist of the unit sphere of a uniformly convex normed space by using the localization technique in codimension greater than one and a strong version of the Borsuk-Ulam theorem. The tools used in this paper follow ideas of Gromov in [Isoperimetry of waists and concentration of maps, Geom. Funct. Anal. 13 (2003), 178-215] and we also include an independent proof of our main theorem which does not rely on Gromov's waist of the sphere. Our waist inequality in codimension one recovers a version of the Gromov-Milman inequality in [Generalisation of the spherical isoperimetric inequality to uniformly convex Banach spaces, Compositio Math. 62 (1987), 263-282].