Minimizing optimal transport for functions with fixed-size nodal sets

TL;DR

This paper characterizes the functions with fixed nodal points that minimize Wasserstein distance between positive and negative parts, providing sharp bounds and exploring the impact of space geometry on these inequalities.

Contribution

It offers a complete solution for the minimization problem on the line and circle, and reveals how geometry affects the sharpness of uncertainty principle inequalities.

Findings

Sharp constants for uncertainty principle inequalities on line and circle.

Optimal bounds depend on the metric measure space geometry.

Non-sharp bounds on star-graphs due to non-branching assumption violation.

Abstract

Consider the class of zero-mean functions with fixed $L^{\infty}$ and $L^1$ norms and exactly $N\in \mathbb{N}$ nodal points. Which functions $f$ minimize $W_p(f_+,f_-)$, the Wasserstein distance between the measures whose densities are the positive and negative parts? We provide a complete solution to this minimization problem on the line and the circle, which provides sharp constants for previously proven ``uncertainty principle''-type inequalities, i.e., lower bounds on $N\cdot W_p (f_+, f_-)$. We further show that, while such inequalities hold in many metric measure spaces, they are no longer sharp when the non-branching assumption is violated; indeed, for metric star-graphs, the optimal lower bound on $W_p(f_+,f_-)$ is not inversely proportional to the size of the nodal set, $N$. Based on similar reductions, we make connections between the analogous problem of minimizing $W_p(f_+,f_-)$ for $f$ defined on $\Omega\subset\mathbb{R}^d$ with an equivalent optimal domain partition problem.