Open problem, Geometry, Applied and computational mathematics, posed 1966

Moving sofa problem

Open

Determine the supremum $A$ (the sofa constant) of the area of a connected planar region $S \subset \mathbb{R}^2$ that can be moved continuously by rigid motions (translations and rotations) around a right-angled corner in an L-shaped hallway of unit width $1$ .

Research frontier as of 2026

As of 2026, the moving sofa problem is widely considered on the verge of complete resolution following Jineon Baek's November 2024 preprint Optimality of Gerver's Sofa (`arXiv:2411.19826`, 119 pages), which is undergoing formal peer review. Prior to Baek's manuscript, the best peer-reviewed bounds were Joseph Gerver's 1992 lower bound $A_G \approx 2.2195316688$ and Yoav Kallus and Dan Romik's 2018 computer-assisted upper bound $A \le 2.37$ . Baek's proof is purely analytical: it establishes that any optimal sofa must be a monotone sofa with total rotation angle $\pi/2$ , proves an injectivity condition on the envelope of corridors, bounds the area from above by a quadratic functional $Q(S)$ via Green's theorem, and uses Brunn–Minkowski theory to show that $Q(S)$ is uniquely maximized by Gerver's sofa.

Best known results

Gerver's 18-piece analytic sofa gives the peer-reviewed lower bound $A \ge A_G \approx 2.2195316688$ (Gerver, 1992).
Kallus and Romik (2018) proved the peer-reviewed upper bound $A \le 2.37$ , while Jineon Baek's 2024 preprint (`arXiv:2411.19826`) claims the sharp equality $A = A_G \approx 2.2195316688$ .

Tools and where they stop

Tool	Achieved	Where it stops
Envelope differential equations and local area variation (Gerver, Romik)	Characterizes locally area-stationary shapes as intersections of rotating hallways and yields explicit analytic curves for Gerver's sofa and Romik's ambidextrous sofa.	Local variational conditions alone cannot rule out the existence of a different topological contact pattern or a non-monotone rotation path with larger area.
Injectivity condition and Brunn–Minkowski concavity (Baek, 2024 preprint)	Upper-bounds the area of any monotone sofa by a quadratic functional $Q(S)$ computed via Green's theorem on an enclosing Jordan curve, showing $Q(S) \le A_G$ .	Awaits completion of journal peer review, and does not yet resolve Conway's ambidextrous sofa problem where the shape must turn both left and right corners.

Open questions

Is Dan Romik's 2018 algebraic sofa of area $\approx 1.644955$ the unique global maximum for Conway's ambidextrous moving sofa problem?

References

Jineon Baek (2024). Optimality of Gerver's Sofa · arXiv:2411.19826 [preprint, not peer-reviewed]
Joseph L. Gerver (1992). On moving a sofa around a corner · DOI:10.1007/BF00181553
Yoav Kallus, Dan Romik (2018). Improved upper bounds in the moving sofa problem · DOI:10.1016/j.aim.2018.10.022 · arXiv:1706.06630
Dan Romik (2018). Differential equations and exact solutions in the moving sofa problem · DOI:10.1080/10586458.2016.1270858 · arXiv:1606.08111