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1. Extensions

    1. Linear dependence of constants in bi-Lipschitz extension theorems


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      Problem 1.1.

      [Leonid Kovalev] Do bi-Lipschitz extension theorems admit linear dependence on constants?
          Every $L$-bi-Lipschitz embedding of ${\mathbb R}$ into ${\mathbb R}^2$ extends to a $2000L$-bi-Lipschitz map of ${\mathbb R}^2$. Is something similar for the circle? Daneri–Pratelli proved that every $L$-bi-Lipschitz embedding of ${\mathbb S}^1$ into ${\mathbb R}^2$ extends to a $CL^4$-bi-Lipschitz map of ${\mathbb R}^2$.
        1. Remark. [Leonid Kovalev] For bi-Lipschitz embeddings of the circle, the answer is yes for extension to the disk, and no for extension to the plane. Reference: Kovalev, L.V., "Optimal extension of Lipschitz embeddings in the plane", Bull. London Math. Soc. 51 (2019), no. 4, 622-632.

            • Dependence on parameters


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              Problem 1.2.

              [Charles Smart] Do canonical Lipschitz extensions exist which depend nicely on parameters?
                  Eva Kopecka (preprint) has constructed a Kirszbraun extension operator which depends continuously on parameters. Other types of dependence on parameters could be considered.

                • Deterministic proof of the Lee–Naor Lipschitz extension theorem

                      The Lee–Naor theorem states that, given a doubling subset $A$ of a metric space $X$ and an $L$-Lipschitz map from $A$ to a Banach space $Y$, there exists a $C(\log m)L$-Lipschitz extension $X\to Y$. Here $m$ is the doubling constant of $A$.

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                  Problem 1.3.

                  [Piotr Hajlasz] Is there a deterministic proof of the Lee–Naor Lipschitz extension theorem giving the sharp dependence on the doubling constant?
                    1. Remark. There are deterministic proofs of similar results by Lang–Schlichenmaier and Brudnyi–Brudnyi.

                        • Absolute Lipschitz retract constructions

                              ${\mathbb Q}_Q(Y)$ denotes the space of unordered $Q$-tuples $y=[y_1,\ldots,y_Q]$ of elements of $Y$ equipped with the metric $d(y,z) = \min \{ \max \{ d(y_j,z_{\sigma(j)}) \} : \sigma \in S_Q \}$, where $S_Q$ denotes the symmetric group on $Q$ letters.

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                          Problem 1.4.

                          [Thierry De Pauw] If $Y$ is an absolute Lipschitz retract, is the same true of the space ${\mathbb Q}_Q(Y)$?
                            1. Remark. [Leonid Kovalev] The space ${\mathbb Q}_Q(Y)$ is similar to the $Q$th symmetric product space; similar questions could be of interest for symmetric products.

                                • Banach space pairs with the Lipschitz extension property


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                                  Problem 1.5.

                                  [Urs Lang] Does the pair $(L^2,L^1)$ have the Lipschitz extension property?
                                      This problem has been asked by Naor and coauthors in several papers.

                                    • Absolute Lipschitz retracts


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                                      Problem 1.6.

                                      [Urs Lang] Are all Hadamard manifolds absolute Lipschitz retracts?

                                        • Lipschitz homotopy groups of the Heisenberg group

                                              ${\mathbb H}^n$ denotes the $n$th Heisenberg group, equipped with the Carnot-Carathéodory metric. $\pi_k^{\scriptstyle{Lip}}(X)$ denotes the $k$th Lipschitz homotopy group of a metric space $X$. It is known that $\pi_n^{\scriptstyle{Lip}}({\mathbb H}^n) \ne 0$ (DeJarnette–Hajlasz–Lukyanenko–Tyson) [arXiv:1109.4641].

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                                          Problem 1.7.

                                          [Piotr Hajlasz and Jeremy Tyson] If $\pi_k({\mathbb S}^n)$ is nontrivial, is the same true for $\pi_k^{\scriptstyle{Lip}}({\mathbb H}^n)$? In particular, is $\pi_3^{\scriptstyle{Lip}}({\mathbb H}^2) \ne 0$?
                                            1. Remark. Solved by Hajlasz, Tyson, Schikorra, Wenger, Young... in HOMOTOPY GROUPS OF SPHERES AND LIPSCHITZ HOMOTOPY GROUPS OF HEISENBERG GROUPS as well as "Lipschitz Homotopy Groups of the Heisenberg Groups", to appear in GAFA (Geometric and Functional Analysis) in February 2014.

                                                • Lipschitz homotopy groups of the Heisenberg group II


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                                                  Problem 1.8.

                                                  [Piotr Hajlasz and Jeremy Tyson] Does $\pi_n^{\scriptstyle{Lip}}({\mathbb H}^n)$ contain torsion elements?
                                                      See problem Heis-Lip for definitions.

                                                      Cite this as: AimPL: Mapping theory in metric spaces, available at http://aimpl.org/mappingmetric.