Isoperimetry of Waists and Concentration of Maps

Let f : Sn → Rk be a continuous map where Sn is the unit n-sphere. Then there exists a point z ∈ Rk such that the spherical n-volumes of the ε-neighbours of the level Yz = f−1(z) ⊂ Sn, denoted Yz + ε ⊂ Sn, satisfy Vol(Yz + ε) ≥ Vol(Sn−k + ε) ( )Sn for all ε > 0, where Sn−k ⊂ Sn denotes an equatorial (n− k)-sphere. This is proven in §5.9. Remarks. (a) If k = n, and card f−1(z) ≤ 2, z ∈ Rn, then ( )Sn applied to ε = π/2 amounts to the Borsuk–Ulam theorem: some level f−1(z) of f : Sn → Rk equals a pair of opposite points. Not surprisingly, our argument in the general case depends on Z2-cohomological considerations. (b) If k = 1 one may take the Levy mean of f for z ∈ R, where the level f−1(z) ⊂ Sn divides the sphere into equal halves (i.e. where Vol(f−1(−∞, z]) and Vol(f−1[z,∞)) are both ≥ 12 Vol(Sn)). Then ( )SN follows from the spherical isoperimetric inequality (see 9.2.B). (c) The inequality ( )Sn for ε → 0 shows that the Minkowski m-volume of f−1(z) for m = n − k is ≥ than that of the equatorial sphere Sm ⊂ Sn. Yet, it remains unclear if some level of f has the Hausdorff measure ≥ Volm(S). (If f is a generic smooth map, a level with Haumesm ≥ Volm(S) is delivered by the Almgren–Morse theory, see [Gr5].)