Rigidity of unit spheres in finite-dimensional real normed spaces

Abstract

We prove that finite-dimensional real normed spaces with isometric unit spheres are linearly isometric, where each sphere carries the distance induced by the ambient norm. This answers the object-level question of Kadets and Martín in finite dimensions, without asserting linear extension of a prescribed sphere isometry. We associate with each norm a family of Plücker contraction bodies generated by volume-normalized maximal minors of linear contractions into finite-dimensional ${\ell }_{\infty }$ spaces. Null-Lagrangian identities show that the sphere metric determines these bodies. A finite exposed-face construction recovers almost-norming contractions from the convexified data, and compactness together with an exact volume identity yields a linear isometry.

23 pages