# Transport and Interface: an Uncertainty Principle for the Wasserstein distance

## TL;DR

This paper establishes a fundamental link between the cost of optimal transport between positive and negative parts of a function and the size of its zero set, revealing an uncertainty principle in Wasserstein geometry.

## Contribution

It introduces a novel uncertainty principle connecting Wasserstein transport cost and nodal set measure, with applications to eigenfunction zero sets in higher dimensions.

## Key findings

- Small Wasserstein distance implies large zero set measure.
- Derived a quantitative inequality relating transport cost and nodal set size.
- Applied the principle to eigenfunctions, showing they cannot have arbitrarily small zero sets.

## Abstract

Let $f: [0,1]^d \rightarrow \mathbb{R}$ 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 $\left\{x \in (0,1)^d: f(x) = 0 \right\}$ 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.