Metric Markov cotype two of ℓ₁

Metric Markov cotype two: moving points on the states of a reversible Markov chain so that neighbouring states carry nearby points, at a cost controlled by the variation of the old points over a time interval. The theorem: the sequence space ℓ₁ has this property, with an explicit constant. Its proof: cuts of finitely many points, a cubic map of a cube that is flat at the vertices, a fourth-power potential, and a walk that stops at a geometric random time; the new points are means of medians of three. Also: Hilbert space, the spaces L₁(μ), complex scalars, and the extension of Lipschitz maps from subsets of Hilbert space into ℓ₁ (from one named hypothesis, the extension theorem of Mendel and Naor). The course is an exposition of OpenAI's proof (2026).

Four lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Metric Markov cotype two
2. Cuts, the cubic map and a quartic potential
3. The stopped walk and the cotype inequality
4. Complex scalars, measure spaces and Lipschitz extension

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Markov type and the reduction to a finite walk (Markov type and superreflexivity)
• The spectral theorem for symmetric matrices (core course Linear Algebra); for the fourth lesson, integrable functions (core course Measure and Integration) and bounded linear functionals (core course Functional Analysis).