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arXiv · 2608.18257

Banach's Isometric Conjecture over the Complex Field

Antonio Acuaviva·Tomasz Kania
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Abstract

We complete Banach's isometric conjecture over the complex field. More precisely, if \(X\) is a complex normed space and, for some \(2\leqslant n<\dim_{\C}X\), all its \(n\)-dimensional complex subspaces are isometric as metric spaces, then the norm is induced by a Hermitian inner product. We also prove the quaternionic counterpart. The central geometric argument first treats real star bodies without convexity or central symmetry; applied to circled complex or quaternionic bodies, it shows that mutually real-linearly equivalent hyperplane sections force the ambient body to be a Hermitian ellipsoid. The proof adapts the bundle-degree mechanism introduced by Lu and Yang for the real case. Finally, we obtain extensions to absolutely homogeneous functions, graded Fr\'echet spaces, metrisable locally convex spaces, and compatible translation-invariant metrics.

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Antonio Acuaviva, Tomasz Kania. 2026-08-18. Banach's Isometric Conjecture over the Complex Field. https://arxiv.org/abs/2608.18257

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