Minimizing Optimal Transport for Functions with Fixed-Size Nodal Sets

Qiang Du, Amir Sagiv*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Article number95
JournalJournal of Nonlinear Science
Issue number5
StatePublished - Oct 2023
Externally publishedYes


  • Metric graph
  • Nodal set
  • Optimal partition
  • Uncertainty principle
  • Wasserstein


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