Metric Markov cotype of ℓ₁

OpenAI's answer (2026) to a question of Mendel and Naor: the space ℓ₁ has metric Markov cotype two. The course compares geometric and uniform averages of time along stationary Markov chains, writes finite subsets of ℓ₁ through weighted cuts, proves a flatness inequality for the cubic 3r²−2r³, and builds the required points from a killed random walk; they turn out to be expected medians of three stopped-walk endpoints.

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

1. Markov chains and metric Markov cotype
2. Cuts and a flat cubic
3. A stopped walk and the cotype of ℓ₁

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Finite Markov chains and geometric series; elementary inequalities (Cauchy–Schwarz and Minkowski).