2021/10/28 by Qiang Du, Amir Sagiv, Du, Qiang +1
Environmental Science · Mathematics · #28A75 #49Q05 #49Q22 #52C35 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Groundwater flow and contamination studies #Markov Chains and Monte Carlo Methods #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2110.14837
openalex publication_date 2021/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider the class of zero-mean functions with fixed L∞ and L1 norms and exactly N∈ ℕ nodal points. Which functions f minimize Wp(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⋅ Wp (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 Wp(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 Wp(f+,f-) for f defined on Ω⊂ℝd with an equivalent optimal domain partition problem.