The finite-dimensional equivalence that fails to extend
Poly and Raby proved in 1984 that for k ≥ 2, a closed set M in a finite-dimensional Euclidean space is a C^k-submanifold near a point x̄ ∈ M if and only if the squared distance function d_M^2 is of class C^k near x̄, with the same order of differentiability. The paper frames the natural open question as whether this equivalence survives in infinite dimensions.
The answer given here is that it does not survive unchanged. The characterization splits into two statements, one for weakly C^k-submanifolds and one for C^k-submanifolds, and the second requires an additional hypothesis beyond smoothness of d_M^2.
Squared distance smoothness alone gives weakly C^k sets
The first half of the split is Theorem 5.3: for a closed set M ⊂ H, k ≥ 2, and x̄ ∈ M, d_M^2 is of class C^k near x̄ if and only if M is a weakly C^k-submanifold near x̄. A weakly C^k-submanifold is defined as a set admitting a local graph representation by a weakly C^k mapping θ, meaning every scalarization y* ∘ θ is C^k even though θ itself need not be.
The mechanism is that C^k smoothness of d_M^2 controls the normal deviation (z, w) ↦ Dθ(z)* w at order C^{k−1}, not θ itself at order C^k. This is equivalent to joint C^k smoothness of (z, w) ↦ ⟨θ(z), w⟩, which is exactly the weak C^k condition. So the natural object that squared-distance smoothness detects is the weak regularity of the graph, not its strong regularity.
A curve where d_M^2 is C^k but M is not
Section 4 constructs, for every k ≥ 2, a closed set M ⊂ ℝ × ℓ^2 containing 0 whose squared distance function d_M^2 is of class C^k near 0, while M is not a C^k-submanifold there. The set is the graph of a curve θ: ℝ → ℓ^2 built from bump functions with disjoint supports I_n = (3r_n/4, 5r_n/4), r_n = 4^{−n}, and decreasing amplitudes r_n^k.
The curve θ is C^{k−1} and weakly C^k, but θ^{(k−1)} is not Fréchet differentiable at 0: along the sequence t_n = r_n, the difference quotient ‖θ^{(k−1)}(r_n) − θ^{(k−1)}(0)‖ / r_n equals 1. Since θ^{(k−1)} vanishes for t ≤ 0, any derivative at 0 would have to be zero, a contradiction. The paper states that this example is tight: it shows the extra condition cannot be omitted, even when the manifold is a curve.
The pointwise equicontinuity condition that restores C^k
The second half of the split is Theorem 6.2. After translating x̄ to 0, M is a C^k-submanifold near 0 if and only if d_M^2 is of class C^k near 0 and, for the Poly–Raby coordinates Ψ, there exist a neighborhood V of 0 in Z = rg(DP_M(0)) and ρ > 0 such that g(z, w) := −(1/2) π_Z ∇d_M^2(Ψ^{−1}(z, w)) is well-defined on V × B_{Z^⊥}(0, ρ) and the family {D_1^{k−1} g(·, w) : w ∈ Z^⊥, ‖w‖ ≤ ρ} is equicontinuous at each z_0 ∈ V.
The paper interprets this condition as pointwise equicontinuity of the last tangential derivative of π_Z ∘ ∇d_M^2, uniform with respect to the normal deviation w ∈ Z^⊥ in Poly–Raby coordinates. It is strictly stronger than plain C^k smoothness of d_M^2, as the counterexample shows, and strictly weaker than local uniform continuity of D^k d_M^2 used in earlier work.
Poly–Raby coordinates remain the common geometric basis
Both characterizations rest on the same Poly–Raby diffeomorphism Ψ(x) = (π_Z(P_M(x)), π_{Z^⊥}(x − P_M(x))), where Z = rg(DP_M(0)) and P_M is the metric projection. Proposition 3.1 shows that when d_M^2 is C^k near 0, Ψ is a C^{k−1}-diffeomorphism, the graph map θ(z) = π_{Z^⊥} Ψ^{−1}(z, 0) is C^{k−1}, and the inverse has the explicit form Ψ^{−1}(z, w) = (z − Dθ(z)* w, θ(z) + w).
The key identity is Dθ(z)* w = −(1/2) π_Z ∇d_M^2(Ψ^{−1}(z, w)). It shows that smoothness of d_M^2 is tied to the joint C^{k−1} smoothness of the normal deviation (z, w) ↦ Dθ(z)* w, not directly to the C^k smoothness of θ. The paper notes that the obstruction to extending Poly and Raby in infinite dimensions is not the construction of Ψ, the graph representation, or the inverse formula, but precisely the step from smoothness of z ↦ Dθ(z)* w for each fixed w to the same smoothness of the operator-valued map z ↦ Dθ(z)*.
Finite codimension collapses the split
Corollary 3.3 records that if d_M^2 is of class C^k near x̄ and ker(DP_M(0)) is finite-dimensional, then M is a C^k-submanifold near x̄ with finite codimension, and conversely. In that regime the compactness argument used by Poly and Raby still works, so the weak and strong characterizations coincide.
The paper's summary table states that conditions 1 to 3 collapse for finite codimension of M. This means the split is a genuinely infinite-dimensional phenomenon: the extra pointwise equicontinuity condition is needed only when the codimension is infinite, and the counterexample lives in ℝ × ℓ^2, which has infinite codimension.
A four-row hierarchy of smoothness characterizations
The paper's final table organizes the results for k ≥ 2 into four rows. d_M^2 ∈ C^k corresponds to M being a weakly C^k-submanifold. d_M^2 ∈ C^k together with condition (6.5) corresponds to M being a C^k-submanifold. d_M^2 ∈ C^{k,0} corresponds to M being a C^{k,0}-submanifold, and d_M^2 ∈ C^{k,1} corresponds to M being a C^{k,1}-submanifold.
The hierarchy makes the trade-off explicit: each step up in the regularity of M requires a strictly stronger hypothesis on d_M^2. The pointwise equicontinuity condition sits between plain C^k smoothness and local uniform continuity of D^k d_M^2, so it is the minimal addition that upgrades the weak conclusion to the strong one.
What to check before applying the theorem
For a reader deciding which conclusion applies, the decisive question is whether the codimension of M is finite or infinite. In finite codimension, C^k smoothness of d_M^2 alone gives a C^k-submanifold, as in the original Poly and Raby theorem. In infinite codimension, the same smoothness gives only a weakly C^k-submanifold, and the pointwise equicontinuity condition (6.5) on the last tangential derivative of π_Z ∘ ∇d_M^2, uniform over the normal deviation w, is what separates the two cases.
The paper's own priority is this split: the weak C^k characterization is the natural home of squared-distance smoothness, and the strong C^k characterization is the weak one plus one equicontinuity hypothesis. The counterexample in ℝ × ℓ^2 shows the hypothesis is not redundant, so any application that needs a genuine C^k-submanifold in infinite codimension must verify condition (6.5) rather than rely on smoothness of d_M^2 alone.
Sources used in this answer
Poly and Raby's theorem splits in infinite dimensions
C^k smoothness of the squared distance function characterizes weakly C^k-submanifolds in infinite-dimensional Hilbert spaces, while C^k-submanifolds require an additional pointwise equicontinuity condition on the highest-order derivative of the squared distance.