A solution to Banach's isometric conjecture

Xinbao Lu, Kaiwen Yang

For odd n≥3, Lu and Yang obtain a global Lipschitz family of exact hyperplane-section isometries after a positive-degree bundle pullback. Combined with their degree argument, this proves that a real Banach space whose n-dimensional subspaces are all linearly isometric is Hilbert when 2≤n<dim X.

Sources for this record

What was already known

Earlier work covered even n, most odd cases and real n=3; odd codimension-one cases remained.

AI involvement AI-centered

Xinbao Lu and Kaiwen Yang reduced the problem to the globalization step. ChatGPT 5.5/5.6 Pro supplied its central approach, GPT-5.6 Sol drafted the section, and the authors checked and rewrote it.

AI supplied the central missing globalization approach within a human reduction and human-completed proof.

Importance Major

The AI-origin globalization step unlocks the remaining real cases of Banach’s isometric conjecture.

Limitations

The AI contribution is the globalization step, not the authors’ final degree argument. The paper does not settle the complex case; a later broader proof is separately credited.

Technical verification Formal (human)

The authors explicitly report checking and rewriting the full AI-drafted Section 3 containing the assessed globalization theorem; the review has not independently repeated that human check.

Independent checking

author check and rewrite of the AI-generated draft

Checker
Xinbao Lu and Kaiwen Yang
Scope
Section 3 globalization argument culminating in Theorem 3.10
Basis
Reported by the linked checker
Evidence source

Human understanding Full

The authors explain the bundle trivialization and degree mechanism; Acuaviva–Kania later adapt it explicitly to a broader setting.

Correction and publication history

2026-10-18 · publication

The assessed result is the real Banach case and its AI-origin globalization step; later extensions are separate.

Read this published revision