Ball's extension problem for Lipschitz maps from Hilbert space into $\ell_1$, via the Mendel-Naor metric Markov cotype question

Theorem 1.1: real $\ell_1$ has metric Markov cotype two with $N_2(\ell_1)\le12\sqrt{21}$, uniformly in the number of points, coordinates and time parameter; complex $\ell_1$ follows with $12\sqrt{42}$. Corollary: there is a universal $K$ such that every Lipschitz $f:S\to\ell_1$, $S$ any subset of a real Hilbert space, extends to $F:H\to\ell_1$ with $\mathrm{Lip}(F)\le K\,\mathrm{Lip}(f)$. The construction is nonlinear (expected coordinatewise medians of geometric-walk endpoints after cut smoothing), consistent with the known failure of Ball's linear Markov cotype for $\ell_1$ and the Kalton-based subspace counterexample. No optimal constant is claimed, and targets other than $\ell_1$ are not treated.

Theorem 1.1: real $\ell_1$ has metric Markov cotype two with $N_2(\ell_1)\le12\sqrt{21}$, uniformly in the number of points, coordinates and time parameter; complex $\ell_1$ follows with $12\sqrt{42}$. Corollary: there is a universal $K$ such that every Lipschitz $f:S\to\ell_1$, $S$ any subset of a real Hilbert space, extends to $F:H\to\ell_1$ with $\mathrm{Lip}(F)\le K\,\mathrm{Lip}(f)$. The construction is nonlinear (expected coordinatewise medians of geometric-walk endpoints after cut smoothing), consistent with the known failure of Ball's linear Markov cotype for $\ell_1$ and the Kalton-based subspace counterexample. No optimal constant is claimed, and targets other than $\ell_1$ are not treated.