366. The planar Mumford–Shah regularity conjecture and local weak-L4 gradient bounds

The planar Mumford–Shah regularity conjecture and local weak-L⁴ gradient bounds

1 paper

Resolves the interior regularity conjecture for reduced absolute planar Mumford–Shah minimizers with bounded fidelity data. Locally, the closed discontinuity set is a \(C^{1,\alpha}\) arc, a regular crack tip, or three arcs meeting at \(120^\circ\); only finitely many global connected components meet any compact interior region.

Papers 1

Interior regularity of planar Mumford–Shah minimizers

OpenAI · September 24, 2026

We prove the interior regularity assertion of the planar Mumford–Shah conjecture for reduced absolute minimizers with bounded fidelity data. Every interior point of the closed discontinuity set has a neighborhood consisting of a \(C^{1,\alpha}\) arc, an arc ending at that point, or three such arcs meeting at 120 degrees. Only finitely many global connected components meet any relatively compact open set.

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@misc{OAI:Interior-regularity-of-planar-Mumford-Shah-minimizers-September-24-2026,
  author = {{OpenAI}},
  title = {{Interior regularity of planar Mumford--Shah minimizers}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Interior-regularity-of-planar-Mumford-Shah-minimizers-September-24-2026/Interior-regularity-of-planar-Mumford-Shah-minimizers-September-24-2026.pdf}{OAI:Interior-regularity-of-planar-Mumford-Shah-minimizers-September-24-2026}},
  year = {2026}
}

The planar Mumford–Shah regularity conjecture and local weak-L⁴ gradient bounds

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Interior regularity of planar Mumford–Shah minimizers

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