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arXiv · 2609.25065

Affine Anchors and Cylinder Obstructions in the Three-Dimensional Tingley Problem

Yicen Ma
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Abstract

Let $X$ be a three-dimensional real Banach space and let $f:S_X\to S_Y$ be a surjective isometry. We study the propagation of an affine formula for $f$ on a relatively open part of $S_X$. A finite family of antipodal distance coordinates propagates a linear anchor except at three explicitly described degeneracies: a facet, an open face-star, or a family of chord cones with a common cylindrical kernel. We prove that, if the norm has no fixed-direction cylindrical open cone, every affine open anchor is linear and global. This yields a source-geometric criterion for the Mazur--Ulam property and covers, among other non-strictly-convex examples, the Euclidean double cone. We also give a segment-saturation criterion which proves the property for every prism $Z\oplus_\infty\mathbb R$, where $Z$ is an arbitrary real Banach plane. Finally, in the remaining cylindrical regime, we prove that a chord-saturated same-kernel network cannot be confined to one proper projective quotient arc. We then show that this multi-arc difficulty and the transverse-ruled part of producing an initial affine anchor reach the same final obstruction: upgrading norm calibration on an ambient open cone, together with calibration on a few spherical segments, to pointwise calibration of the sphere map on a two-dimensional open patch.

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Yicen Ma. 2026-09-15. Affine Anchors and Cylinder Obstructions in the Three-Dimensional Tingley Problem. https://arxiv.org/abs/2609.25065

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