Affine Anchors and Cylinder Obstructions in the Three-Dimensional Tingley Problem
Abstract
Let X be a three-dimensional real Banach space and let f:SX→ SY be a surjective isometry. We study the propagation of an affine formula for f on a relatively open part of SX. 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⊕∞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.
Related
- Tingley's Problem for Haagerup Noncommutative Lp-Spaces, 1<p≠ 2<∞
- Isometries of Unit Spheres of SFS-Spaces and a Geometric Approach to Tingley's Problem
- Obstructions in Symplectic Embedding Problems for Four-Dimensional Convex Toric Domains
- Lazard's Realization Problem for N-Series: Bar Obstructions and Finite-Quotient Descent
- The Bernstein problem in higher dimensional Heisenberg groups
BibTeX
@misc{ma2026affine,
title = {{Affine Anchors and Cylinder Obstructions in the Three-Dimensional Tingley Problem}},
author = {Yicen Ma},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25065},
eprint = {2609.25065},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25065},
}
RIS
TY - UNPB
TI - Affine Anchors and Cylinder Obstructions in the Three-Dimensional Tingley Problem
AU - Yicen Ma
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.25065
UR - https://arxiv.org/abs/2609.25065
AB - Let X be a three-dimensional real Banach space and let f:S_X→ 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⊕_∞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.
ER -