Minimizing Optimal Transport for Functions with Fixed-Size Nodal Sets

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