Minimizing Optimal Transport for Functions with Fixed-Size Nodal Sets

Consider the class of zero-mean functions with fixed L^∞ and L¹ norms and exactly N∈ 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· 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 Ω ⊂ R^d with an equivalent optimal domain partition problem.