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

Ball (1992) introduced Markov type and cotype to study Lipschitz extension and asked whether every Lipschitz map $f$ from an arbitrary subset of a Hilbert space $H$ into $L_1$ extends to all of $H$ with $\mathrm{Lip}(F)\le K\,\mathrm{Lip}(f)$ for a universal $K$. Mendel and Naor (2013) formulated metric Markov cotype two in Cesaro form: $X$ has it with constant $C$ if for every reversible stochastic matrix $A$ with stationary $\pi$, every $t$ and points $x_1,\dots,x_n\in X$ there are $y_1,\dots,y_n\in X$ with $\sum_i\pi_i d(x_i,y_i)^2+t\sum_{i,j}\pi_i a_{ij}d(y_i,y_j)^2\le C^2\sum_{i,j}\pi_i(\frac1t\sum_{s=1}^tA^s)_{ij}d(x_i,x_j)^2$. They showed that a positive answer for $\ell_1$ yields the extension, and asked (Question 1.15): is $N_2(\ell_1)<\infty$, that is, does $\ell_1$ have metric Markov cotype two?

Result

Proved(see note)

Status

Candidate (review pending)

AI contribution

AI-discovered

Method

Argument

Field

Metric geometry; nonlinear Banach space theory, Lipschitz extension

Posed by

Keith Ball (1992, extension problem); Manor Mendel and Assaf Naor (2013, Question 1.15)

Year posed

1992

Years open

34y

Solved

2026-10-05

Model

Unreleased internal OpenAI model

Vendor

OpenAI

Collaborators

—

Verification

Unreviewed

Publication

Announced

Collection

OpenAI math release (October 2026), version adc7f12

Significance

35 / 100*

Disclosed cost

—

Wikipedia

No dedicated article