A power saving for planar halving lines
OpenAI's theorem (2026) that n points in the plane with no three on a line have at most C n^(4/3 - eps) halving lines for an absolute eps > 0, the first improvement of Dey's exponent 4/3; the same bound for the switches of rank sweeps uniformly in the rank, and the bound C n (k+1)^(1/3 - eps0) for k-sets. Dey's bound through convex chains and crossings, the induction from a packing estimate to a power saving, and the packing estimate by contradiction: rank coordinates, weak limits of measures, monotone functions of two variables, three transverse foliations, a zoom with radial comparison, a spatial measure and a count of boundary activity.
Thirteen lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.
- Halving lines and rank sweeps
- Chains, crossings and Dey's bound
- From packing to a power saving
- Measures on compact metric spaces
- Rank coordinates
- Limiting measures
- Monotone functions of two variables
- Independent gradients and tangent directions
- Three transverse foliations
- The zoom at a rank point
- A spatial measure
- Boundary activity
- Levels and k-sets
Theorems and assumptions · Reader and editable sources · Course record
Prerequisites
- Haar measure on locally compact groups (Distributions, kernels and analytic singularities)
- Drawings and crossing numbers (Crossing numbers of complete and complete bipartite graphs)
- Real analysis in one and several variables (core courses Real Analysis I and II).
- Measure theory: the Radon–Nikodym theorem, the Lebesgue decomposition and Rademacher's theorem (core course Measure and Integration).
- The Riesz representation theorem (lesson Haar measure on locally compact groups) and Brouwer's fixed-point theorem (core course Algebraic Topology).
- Planar graphs with v ≥ 3 vertices have at most 3v − 6 edges (core course Applied Combinatorics).