Erratum: A coordinate-free proof of the finiteness principle for Whitney's extension problem (Revista Matematica Iberoamericana (2020) 36:7 (1917-1956) DOI: 10.4171/rmi/1186)

The purpose of this note is to draw attention to a misleading remark in the introduction of [1]. In our discussion of Theorem 1.2 we make the following claim: “one may check that the constants λ₁ and λ₂ in Theorem 1.2 are harmless polynomial functions of D”. Although we believe this to be true, the statement does not follow from the arguments of the paper. Several modifications are needed to obtain the claim, which we will now describe. The first issue relates to the ineffective constant R₀ in Lemma 3.13 which arises due to the use of a compactness argument in the proof. This issue can be resolved, but the proofs are not as straightforward as we had once thought. In a forthcoming paper by the first three named authors, we give a direct geometric proof of Lemma 3.13 with the constant R₀ = O(exp(poly(D))). This is sufficient to obtain the polynomial dependence of λ₁ and λ₂, as claimed. The second issue relates to an unfortunate typo appearing in Section 8 of the paper: in Lemma 8.7, the constant C should be replaced by C · C_old; here, C_old = C^#(K−1), and C is a constant determined by m and n. When accounting for the missing factor of C_old, we find that a number of constants in Sections 8 and 9 which are claimed to depend only on m, n, actually depend also on the inductive parameter K. In Section 9.1 we claim that C^#(K) and ℓ^#(K) have the form (0.1) C^#(K) = C^K, ℓ^#(K) = χ · K, where C and χ are constants determined by m and n. The scaling (0.1) is responsible for the scaling C^# = exp(λ₁C(E)) and k^# = exp(λ₂C(E)) in Theorem 1.2; see Remark 5.7. In the rest of this note, we will demonstrate that the scaling (0.1) is not ruined when we properly account for the factor of C_old in Lemma 8.7. We first state an amended form of Lemma 8.1, which is the main result of Section 8 of [1]. We then explain why the amended Lemma 8.1 is sufficient to obtain the scaling (0.1). We finally discuss the proof of the amended Lemma 8.1.