On the behavior of 1-Laplacian ratio cuts on nearly rectangular domains

Wesley Hamilton, Jeremy L. Marzuola, Hau Tieng Wu

Research output: Contribution to journalArticlepeer-review


The p-Laplacian has attracted more and more attention in data analysis disciplines in the past decade. However, there is still a knowledge gap about its behavior, which limits its practical application. In this paper, we are interested in its iterative behavior in domains contained in two-dimensional Euclidean space. Given a connected set Ω0 ⊂ R2, define a sequence of sets (Ωn)n=0 where Ωn+1 is the subset of Ωn where the first eigenfunction of the (properly normalized) Neumann p-Laplacian −Δ(p)φ = λ1|φ|p−2φ is positive (or negative). For p = 1, this is also referred to as the ratio cut of the domain. We conjecture that these sets converge to the set of rectangles with eccentricity bounded by 2 in the Gromov–Hausdorff distance as long as they have a certain distance to the boundary ∂Ω0. We establish some aspects of this conjecture for p = 1 where we prove that (1) the 1-Laplacian spectral cut of domains sufficiently close to rectangles is a circular arc that is closer to flat than the original domain (leading eventually to quadrilaterals) and (2) quadrilaterals close to a rectangle of aspect ratio 2 stay close to quadrilaterals and move closer to rectangles in a suitable metric. We also discuss some numerical aspects and pose many open questions.

Original languageEnglish (US)
Pages (from-to)1563-1610
Number of pages48
JournalInformation and Inference
Issue number4
StatePublished - Dec 1 2021


  • domain deformation
  • ratio cuts

ASJC Scopus subject areas

  • Analysis
  • Statistics and Probability
  • Numerical Analysis
  • Computational Theory and Mathematics
  • Applied Mathematics


Dive into the research topics of 'On the behavior of 1-Laplacian ratio cuts on nearly rectangular domains'. Together they form a unique fingerprint.

Cite this