An upper bound for the three-dimensional moving sofa problem in a snaky corridor
Abstract
Let KM ⊂ ℝ3 be the "snaky" corridor with unit-square cross-section: an infinite entry arm parallel to the x-axis, a middle arm parallel to the y-axis whose clear straight section has length M, and an infinite exit arm parallel to the z-axis, with the two turns lying in the x-y and y-z planes. Let V*(KM) be the supremum of the volumes of connected bodies that can be moved continuously (by rigid motions) from the entry arm to the exit arm while remaining inside KM. We prove V*(KM) ≤ 7.8691 for every M ≥ 3.53 (Theorem A); by a sharper analysis of the same mechanism, V*(KM) ≤ 4.2862 for every M ≥ 2.61 (Theorem B); and, by replacing the final analytic caps with a certified branch-and-bound computation over the compact family of entry tubes compatible with the forced events, V*(KM) ≤ 3.4083 for every M > 2.2 (Theorem C; computer-assisted, in double precision with conservative slack). In particular all three bounds apply to the corridor with M = 4 used in Problem 6.64 of Georgiev, Gómez-Serrano, Tao and Wagner (arXiv:2511.02864), and uniformly to every "turns far apart" reading of that problem. To the best of our knowledge, this is the first upper bound for the snaky-corridor 3D moving sofa problem; previously not even finiteness of V* had been recorded. This note proves upper bounds only; combined with the certified lower bound 1.8558404 from the author's separate certification work via the problem's own evaluation pipeline, the state of the problem becomes 1.8558404 ≤ V*(K4) ≤ 3.4083. Theorems A and B are elementary and self-contained: a first-crossing (elbow) event forces a definite tilt of the body's own entry tube, and Fubini-type chord estimates for tilted tubes cap the volume. Theorem C certifies, by the same forced-event skeleton, the largest corridor volume that any admissible entry tube can carry.
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BibTeX
@misc{tang2026upper,
title = {{An upper bound for the three-dimensional moving sofa problem in a snaky corridor}},
author = {Yuzhe Tang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14050},
eprint = {2610.14050},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14050},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - An upper bound for the three-dimensional moving sofa problem in a snaky corridor
AU - Yuzhe Tang
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.14050
UR - https://arxiv.org/abs/2610.14050
AB - Let K_M ⊂ ℝ^3 be the "snaky" corridor with unit-square cross-section: an infinite entry arm parallel to the x-axis, a middle arm parallel to the y-axis whose clear straight section has length M, and an infinite exit arm parallel to the z-axis, with the two turns lying in the x-y and y-z planes. Let V^*(K_M) be the supremum of the volumes of connected bodies that can be moved continuously (by rigid motions) from the entry arm to the exit arm while remaining inside K_M. We prove V^*(K_M) ≤ 7.8691 for every M ≥ 3.53 (Theorem A); by a sharper analysis of the same mechanism, V^*(K_M) ≤ 4.2862 for every M ≥ 2.61 (Theorem B); and, by replacing the final analytic caps with a certified branch-and-bound computation over the compact family of entry tubes compatible with the forced events, V^*(K_M) ≤ 3.4083 for every M > 2.2 (Theorem C; computer-assisted, in double precision with conservative slack). In particular all three bounds apply to the corridor with M = 4 used in Problem 6.64 of Georgiev, Gómez-Serrano, Tao and Wagner (arXiv:2511.02864), and uniformly to every "turns far apart" reading of that problem. To the best of our knowledge, this is the first upper bound for the snaky-corridor 3D moving sofa problem; previously not even finiteness of V^* had been recorded. This note proves upper bounds only; combined with the certified lower bound 1.8558404 from the author's separate certification work via the problem's own evaluation pipeline, the state of the problem becomes 1.8558404 ≤ V^*(K_4) ≤ 3.4083. Theorems A and B are elementary and self-contained: a first-crossing (elbow) event forces a definite tilt of the body's own entry tube, and Fubini-type chord estimates for tilted tubes cap the volume. Theorem C certifies, by the same forced-event skeleton, the largest corridor volume that any admissible entry tube can carry.
ER -