Selected research history · publication v4

Schiffer and Pompeiu conjectures

Follow the ideas, results and limitations over time. This is a selected history, not a complete literature survey or a measure of how close the problem is to being solved.

Current state & remaining question →

Viewing the timeline from pinned publication v4.

4 selected milestones · Source coverage checked 2026-10-13 · Historical annotation and mathematical review remain separate.

1929 publication
Foundation
Pompeiu asks whether domain averages determine a function
The cited papers trace the integral-geometric question to Pompeiu.
Scope & review: Topology and boundary regularity must be fixed when stating the disk-rigidity conjecture.
Statements & sources
- The planar Pompeiu property question →
- A computer-assisted counterexample to the planar Pompeiu and Schiffer conjectures ↗Main theorem, computational certificate appendix and AI declaration
- Counterexamples to Schiffer’s Conjecture ↗Theorem 1.1, Corollary 1.2 and AI/Lean acknowledgement
Historical annotation review: pending. The date refers to the publication, with the precision supported by the source.
Aug 3, 2026 preprint
Related result
An analytic computer-assisted construction
Colbrook and Stepaniants publish the tenfold counterexample and a certificate archive.
Scope & review: Rigorous interval and tail estimates certify an exact domain; a floating-point drawing alone would not do so.
Statements & sources
- An analytic tenfold counterexample →
- A computer-assisted counterexample to the planar Pompeiu and Schiffer conjectures ↗Main theorem, computational certificate appendix and AI declaration
- Numerical certificate and reproduction report ↗REPRODUCTION_REPORT.md; validation archive; Zenodo DOI 10.5281/zenodo.21765287
Historical annotation review: pending. The date refers to the preprint, with the precision supported by the source.
Aug 5, 2026 preprint
Related result
An infinite family by uniform bifurcation
Cao-Labora and de Dios Pont publish smooth star-shaped counterexamples for infinitely many symmetry orders.
Scope & review: The linked earlier-draft Lean statement asserts C² regularity and follows a modified analytic path.
Statements & sources
- Infinitely many smooth symmetry orders →
- The normalized C² formal statement →
- Counterexamples to Schiffer’s Conjecture ↗Theorem 1.1, Corollary 1.2 and AI/Lean acknowledgement
- Lean proof of the star-shaped Schiffer/Pompeiu construction ↗Schiffer.Comparator.Schiffer_Star_Shaped and Pompeiu_Star_Shaped; simply-connected corollaries
Historical annotation review: pending. The date refers to the preprint, with the precision supported by the source.
Sep 12, 2026 formalization
Formalization
The analytic certificate receives a public Lean replay
The verification record reports a standard-kernel statement comparison after a complete Lean build.
Scope & review: This formal project belongs to Colbrook–Stepaniants’ analytic construction, separately from de Dios Pont’s C² bifurcation project.
Statements & sources
- An analytic tenfold counterexample →
- Lean proof of the analytic tenfold construction ↗PompeiuSchiffer.Numerics.UnconditionalCounterexamples.exists_common_domain; Comparator definitions and verification record
Historical annotation review: pending. The date refers to the formalization, with the precision supported by the source.

← Back to the Problem

Pompeiu asks whether domain averages determine a function

An analytic computer-assisted construction

An infinite family by uniform bifurcation

The analytic certificate receives a public Lean replay