Say f : [0,1]^3 —> R^2 is a continuous map from the unit cube to the plane. Unsolved problem (I think): ----- Prove there is some point p in R^2 whose inverse image f^(-1)(p) has area >= 1. ----- Here "area" means 2-dimensional Hausdorff measure.* This should properly be called a conjecture. It is related to an apparently difficult — but proven — theorem in geometry, the "waist inequality" of Gromov: Let S^3 be the unit sphere in 4-space, and let g : S^3 —> R^2 be any continuous map, where D^3 is the closed unit ball in R^3. ----- Then there exists some point p of R^2 whose inverse image g^(-1)(p) has area at least the area of the equator of S^3. That is, >= 4π^2, since the equator is a unit 2-sphere). ----- —Dan ————— * Hausdorff r-measure is an ingenious way to assign a reasonable meaning to the notion of r-dimensional measure of a compact subset K of Euclidean space R^N, for any positive real number r. When r is an integer it works like this: Let alph(N) > 0 be the constant for which the volume of an N-dimensional ball D_R of radius R in R^n is equal to vol(D_R) = alpha(N) R^N. Now in R^N, let C be any covering of K by open balls of radii <= eps, for some eps > 0. Define |C|_(r, eps) = alph(N) * (the sum of the r'th power of each radius of a ball in C). Finally, define |C|_r = inf |C|_(r, eps) eps > 0 It's a bit complicated, but it's exactly the right definition.