A solution to Banach conjecture

In this paper, we begin by constructing two families of origin-symmetric star bodies derived from the John ellipsoid of the sections. We then apply the level-set method to obtain a complete proof of Banach’s isometric subspace problem in finite-dimensional spaces via the John ellipsoid.

💡 Research Summary

The paper addresses the long‑standing Banach conjecture, which asks whether a normed vector space (V,‖·‖) in which all n‑dimensional linear subspaces are mutually isometric must necessarily be Euclidean (i.e., induced by an inner product). Geometrically this translates to the statement that an origin‑symmetric convex body K⊂ℝⁿ whose (n‑1)‑dimensional central sections are all linearly equivalent must be an ellipsoid. The author reduces the problem to the case dim V=n+1 and fixes a reference direction ξ₀∈S^{n‑1}.

The main result, Theorem 1.1, asserts that under the hypothesis that for every direction ξ there exists a linear map ϕ_ξ∈GL(n) satisfying K∩ξ⊥=ϕ_ξ(K∩ξ₀⊥) and ϕ_ξ(ξ₀)=ξ, the body K must be an ellipsoid. The proof proceeds in three conceptual steps.

1. John ellipsoid construction. For each hyperplane section K∩ξ₀⊥ the John ellipsoid J(K∩ξ₀⊥) – the unique maximal‑volume ellipsoid contained in the section – is selected. Because John ellipsoids are equivariant under linear maps, the family {ϕ_ξ(J(K∩ξ₀⊥))} inherits the same linear equivalence as the sections themselves. Using these ellipsoids the author defines two families of origin‑symmetric star bodies:
   • E_α = ⋂_{ξ∈S^{n‑1}} ϕ_ξ(J(K∩ξ₀⊥)) ∩ α·(K∩ξ₀⊥),
   • ˜E_β = ⋃_{ξ∈S^{n‑1}} ϕ_ξ(J(K∩ξ₀⊥)) ∪ β·(K∩ξ₀⊥),
     where α,β∈