The Moving Sofa Problem

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

In $\mathbb R^2$, set \[ H=(-\infty,1]\times[0,1],\qquad V=[0,1]\times(-\infty,1],\qquad L=H\cup V. \] Thus $L$ is a right-angled hallway with both arms of width one. Write $\lambda_2$ for planar Lebesgue measure. A moving sofa is a nonempty connected closed set $S\subseteq\mathbb R^2$ for which there are continuous functions $\theta:[0,1]\to\mathbb R$ and $b:[0,1]\to\mathbb R^2$ such that $\theta(0)=0$ and, with \[ R_\theta=\begin{pmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{pmatrix}, \qquad g_t(x)=R_{\theta(t)}x+b(t), \] one has $g_0(S)\subseteq H$, $g_t(S)\subseteq L$ for all $t\in[0,1]$, and $g_1(S)\subseteq V$. These are continuous orientation-preserving rigid motions; the sofa cannot be deformed or reflected during the motion. Closed sets are Borel measurable. [S2]

Statement recorded in the supplied source.

Determine the moving sofa constant \[ \alpha_{\max}=\sup\{\lambda_2(S): S\text{ is a moving sofa as defined above}\}, \] and describe a sofa of maximum area. The optimization allows every such connected shape and every continuous rigid-motion path through the unit-width right angle. [S2]

Short English statement

Find the greatest area of a single rigid planar shape that can be carried continuously around a right angle in a hallway of width one, and identify a shape attaining it.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 33 (GEO-004). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Jineon Baek, Optimality of Gerver’s Sofa, arXiv:2411.19826v1 (2024), Definitions 1.1.1--1.1.3 and Theorem 1.1.1. https://arxiv.org/html/2411.19826v1.