math.MG — Metric Geometry

A generalized Gerver sofa for angled corridors

Contributed by Henrik Schou Guttesen

The moving sofa problem asks for the maximal area of a two-dimensional object capable of being moved through a right-angled corridor of unit width. J.~L.~Gerver constructed a sofa of area $2.2195\dots$, which was only recently proven to be optimal by J.~Back. Here, I consider the moving sofa problem in an angled corridor, $C_{\varphi}$, of unit width for turn angles $0 < \varphi \le \frac{\pi}{2}$. I first generalize Hammersley's arguments to devise simple analytical lower and upper area bounds for all $0 < \varphi \le \frac{\pi}{2}$. ChatGPT-6 Astra (Codex) is prompted to derive a generalized Gerver sofa parametrized by $\varphi$. The generalized Gerver sofa area is non-analytic at the unique root $\varphi_*\in(0,\pi/2)$ of $\exp({\varphi_*\cot\varphi_*}) = 1+2\cos\varphi_*-\cos^2\varphi_*$. For turn angles smaller than $\varphi_*$ a closed-form expression of the sofa area is obtained. The generalized Gerver sofa is conjectured to be the solution to the moving sofa problem for turn angles $0 < \varphi \le \frac{\pi}{2}$.