Transport and Interface: an Uncertainty Principle for the Wasserstein distance

Let $f:[0,1{]}^d\to \mathbb{R}$ 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 $W^1$), then the nodal set $\{x\in (0,1{)}^d:f(x)=0\}$ has to be large (`if it is always easy to buy milk, there must be many supermarkets’). More precisely, we show that $$ W_1(f_+, f_-) \cdot $\mathcal{H}$^{d-1}\left{x \in (0,1)^d: f(x) = 0 \right} \gtrsim_{d} \left( \frac{|f|{L^1}}{|f|{L^{\infty}}} \right)^{4

• \frac1d} |f|_{L^1} , .$$ 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.