PompeiuProblem.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

9 declarations, 3 contain sorry, 0 declared as axioms

The Pompeiu problem

Let $\Omega \subset \mathbb{R}^{N+1}$ be a bounded domain. A rigid motion is an isometry of $\mathbb{R}^{N+1}$ (a composition of translations and rotations), modelled here by an AffineIsometryEquiv of Euclidean space with itself.

The domain $\Omega$ has the Pompeiu property if the only continuous function $f : \mathbb{R}^{N+1} \to \mathbb{R}$ with $\int_{\sigma(\Omega)} f = 0$ for every rigid motion $\sigma$ is $f \equiv 0$.

The Pompeiu problem (Pompeiu's conjecture) asserted that, among bounded simply connected domains with Lipschitz boundary, the Euclidean ball is the only domain that fails to have the Pompeiu property. That a ball fails is classical: radial functions built from Bessel functions have vanishing integral over every congruent ball. The converse is false. Two independent proofs of this negative conclusion are referenced below.

References:

Wikipedia
Encyclopedia of Mathematics
[BST73] Brown, L., Schreiber, B. M. and Taylor, B. A., Spectral synthesis and the Pompeiu problem, Ann. Inst. Fourier 23 (1973), 125–154. (Proves a ball fails the Pompeiu property; for a ball of radius R the witness is f(x) = sin(a x₁) with J_{n/2}(a R) = 0.)
Zalcman, L., A bibliographic survey of the Pompeiu problem, in Approximation by Solutions of Partial Differential Equations, NATO ASI Ser. 365 (1992), 185–194.
[CLD26] Cao-Labora, G. and de Dios Pont, J., Schiffer, a Lean 4 formalization of a non-disk planar counterexample.
[CS26] Colbrook, M. J. and Stepaniants, G., A computer-assisted counterexample to the planar Pompeiu and Schiffer conjectures, arXiv:2608.01579 (2026).

Imports (1)

FormalConjecturesUtil

abbrev PompeiuProblem.RigidMotion A rigid motion of $\mathbb{R}^{N+1}$: an affine isometry of Euclidean space onto itself (a composition of translations and rotations/reflections) line 61 source
Source documentation
A rigid motion of $\mathbb{R}^{N+1}$: an affine isometry of Euclidean space onto itself (a composition of translations and rotations/reflections).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: AffineIsometryEquiv, Real, EuclideanSpace, Fin, HAdd.hAdd, OfNat.ofNat, ENNReal
24 instances, coercions and auxiliary constants left out.
```
abbrev RigidMotion (N : ℕ) :=
…
```
def PompeiuProblem.HasPompeiuProperty A set Ω has the Pompeiu property if the only continuous f : ℝ^{N+1} → ℝ whose integral over every rigid-motion image σ(Ω) vanishes is the zero function line 66 source
Source documentation
A set Ω has the Pompeiu property if the only continuous f : ℝ^{N+1} → ℝ whose integral over every rigid-motion image σ(Ω) vanishes is the zero function.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set, EuclideanSpace, Real, Fin, HAdd.hAdd, OfNat.ofNat
Proof uses: Continuous, ENNReal, PompeiuProblem.RigidMotion, Eq, MeasureTheory.integral, MeasureTheory.Measure.restrict, MeasureTheory.MeasureSpace.volume, Set.image
46 instances, coercions and auxiliary constants left out.
```
def HasPompeiuProperty (Ω : Set (EuclideanSpace ℝ (Fin (N + 1)))) : Prop :=
…
```
def PompeiuProblem.IsBall Ω is a Euclidean ball: an open metric ball of positive radius line 71 source
Source documentation
Ω is a Euclidean ball: an open metric ball of positive radius.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set, EuclideanSpace, Real, Fin, HAdd.hAdd, OfNat.ofNat
Proof uses: Exists, And, LT.lt, Eq, Metric.ball, ENNReal
18 instances, coercions and auxiliary constants left out.
```
def IsBall (Ω : Set (EuclideanSpace ℝ (Fin (N + 1)))) : Prop :=
…
```
def PompeiuProblem.HasLipschitzBoundary Documented declaration: HasLipschitzBoundary line 79 source
Source documentation
Ω has Lipschitz boundary if every boundary point p has a neighbourhood U in which, after some rigid motion σ (used to choose a "vertical" axis), Ω coincides with the strict subgraph of a Lipschitz function g of the first N coordinates: a point x ∈ U lies in Ω iff its last coordinate (in the σ-frame) is below g applied to its first N coordinates. This is the standard definition of a Lipschitz domain.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set, EuclideanSpace, Real, Fin, HAdd.hAdd, OfNat.ofNat
Proof uses: Membership.mem, frontier, ENNReal, Exists, And, Filter, nhds, PompeiuProblem.RigidMotion, NNReal, LipschitzWith, Iff, LT.lt, Fin.last, WithLp.ofLp, Equiv, WithLp, Equiv.symm, WithLp.equiv, Fin.castSucc
42 instances, coercions and auxiliary constants left out.
```
def HasLipschitzBoundary (Ω : Set (EuclideanSpace ℝ (Fin (N + 1)))) : Prop :=
…
```
structure PompeiuProblem.IsAdmissibleDomain The domain hypotheses of Pompeiu's problem line 90 source
Source documentation
The domain hypotheses of Pompeiu's problem: $\Omega$ is bounded, open, connected (a "domain"), simply connected, has Lipschitz boundary, and for $N \ge 2$ the complement of its closure is connected. For $N = 1$, connectedness of $(\overline{\Omega})^c$ follows from simple connectivity.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set, EuclideanSpace, Real, Fin, HAdd.hAdd, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.
```
structure IsAdmissibleDomain (Ω : Set (EuclideanSpace ℝ (Fin (N + 1)))) : Prop where
…
```
theorem PompeiuProblem.pompeiu_conjecture sorry line 108 source
depends on sorry what this means
Statement uses: Not, Set, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat, PompeiuProblem.IsAdmissibleDomain, Iff, PompeiuProblem.HasPompeiuProperty, PompeiuProblem.IsBall
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem PompeiuProblem.ball_not_hasPompeiuProperty The classical (easy) direction, proved by Brown–Schreiber–Taylor [BST73]: a Euclidean ball fails to have the Pompeiu property sorry line 120 source
Source documentation
The classical (easy) direction, proved by Brown–Schreiber–Taylor [BST73]: a Euclidean ball fails to have the Pompeiu property. For a ball of radius R, an explicit witness is f(x) = sin(a x₁) where a > 0 is chosen so that the Bessel function J_{n/2}(a R) = 0; its integral over every congruent ball vanishes.

depends on sorry what this means
Statement uses: Nat, Set, EuclideanSpace, Real, Fin, HAdd.hAdd, OfNat.ofNat, PompeiuProblem.IsBall, Not, PompeiuProblem.HasPompeiuProperty
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem ball_not_hasPompeiuProperty (Ω : Set (EuclideanSpace ℝ (Fin (N + 1))))
    (hball : IsBall Ω) :
    ¬ HasPompeiuProperty Ω := by
…
```
Mentions: PompeiuProblem.IsBall, PompeiuProblem.HasPompeiuProperty
theorem PompeiuProblem.not_hasPompeiuProperty_imp_ball sorry line 132 source
depends on sorry what this means
Statement uses: Not, Set, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat, PompeiuProblem.IsAdmissibleDomain, PompeiuProblem.HasPompeiuProperty, PompeiuProblem.IsBall
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem PompeiuProblem.integral_zero_eq_zero Documented declaration: integral_zero_eq_zero line 140 source
Source documentation
Sanity check: the zero function always integrates to 0 over every rigid-motion image, so the hypothesis in HasPompeiuProperty is satisfiable (the property is not vacuously false).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set, EuclideanSpace, Real, Fin, HAdd.hAdd, OfNat.ofNat, PompeiuProblem.RigidMotion, Eq, MeasureTheory.integral, ENNReal, MeasureTheory.Measure.restrict, MeasureTheory.MeasureSpace.volume, Set.image
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, MeasureTheory.integral_zero, eq_self
132 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem integral_zero_eq_zero (Ω : Set (EuclideanSpace ℝ (Fin (N + 1)))) (σ : RigidMotion N) :
    (∫ _x in Set.image σ Ω, (0 : ℝ)) = 0 := by
…
```
Mentions: PompeiuProblem.RigidMotion