Transport and interface: An uncertainty principle for the wasserstein distance

Let f : (0, 1)d → ℝ be a continuous function with zero mean and interpret f+ = max(f, 0) and f- = - min(f, 0) as the densities of two measures. We prove that if the cost of transport from f+ to f- is small, in terms of the Wasserstein distance W1(f+, f-), then the Hausdorff measure of the nodal set { x ∈ (0, 1)d : f(x) = 0 has to be large ("if it is always easy to buy milk, there must be many supermarkets"). More precisely, we show that the product of the (d - 1)-dimensional volume of the zero set and the Wasserstein transport cost can be bounded from below in terms of the Lp norms of f. We apply this "uncertainty principle"to the metric Sturm-Liouville theory in higher dimensions to show that a linear combination of eigenfunctions of an elliptic operator cannot have an arbitrarily small zero set.