banach_mazur_distance_conjecture

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Banach-Mazur distance: Maximum distance between n-dimensional Banach spaces. John's theorem gives max d(X,l₂ⁿ) = √n. The maximum d(X,Y) over all pairs is n, achieved by (l₁ⁿ, l∞ⁿ). Bounds on Banach-Mazur compactum diameter are open.

Why this node was retired

The posted statement is

import Mathlib

theorem banach_mazur_distance_conjecture (n : ℕ) (hn : 2 ≤ n)
    (X Y : Type*) [NormedAddCommGroup X] [NormedSpace ℝ X]
    [NormedAddCommGroup Y] [NormedSpace ℝ Y]
    (hX : Module.finrank ℝ X = n) (hY : Module.finrank ℝ Y = n) :
    ∃ (T : X →L[ℝ] Y) (_ : Function.Bijective T),
      ‖T‖ * ‖T.inverse‖ ≤ Real.sqrt n := by
  sorry

The claimed √n bound is for arbitrary pairs of n-dimensional normed spaces; a square and regular hexagon in dimension two have distance≥3/2>√2.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

For arbitrary n-dimensional real normed spaces use the general Banach–Mazur upper bound n. For the √n John ellipsoid bound, require one of the two spaces to be Euclidean.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.