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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 densitiesare the positive and negative parts? We provide a complete solution to thisminimization problem on the line and the circle, which provides sharp constantsfor previously proven ``uncertainty principle''-type inequalities, i.e., lowerbounds on $N\cdot W_p (f_+, f_-)$. We further show that, while suchinequalities hold in many metric measure spaces, they are no longer sharp whenthe non-branching assumption is violated; indeed, for metric star-graphs, theoptimal lower bound on $W_p(f_+,f_-)$ is not inversely proportional to the sizeof the nodal set, $N$. Based on similar reductions, we make connections betweenthe analogous problem of minimizing $W_p(f_+,f_-)$ for $f$ defined on$\Omega\subset\mathbb{R}^d$ with an equivalent optimal domain partitionproblem.

Minimizing optimal transport for functions with fixed-size nodal sets