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Moving Sofa Problem

The Moving Sofa Problem is an open problem in planar geometry and optimal control that seeks the maximal area of a connected, rigid two-dimensional shape that can be maneuvered (translated and rotated) through an L-shaped corridor of unit width without self-intersection.

Formal Statement

Let \(\mathcal C\) denote the infinite L-shaped corridor of unit width:

\[ \mathcal C = \left\{(x,y)\in\mathbb R^2 : 0\le x\le 1,\; y\ge 0\right\}\cup\left\{(x,y)\in\mathbb R^2 : x\ge 0,\; 0\le y\le 1\right\} \]

Find the supremum of areas

\[ A_{\max}= \sup\bigl\{\operatorname{Area}(S)\bigr\} \]

over all connected compact sets \(S\subset\mathbb R^2\) for which there exists a continuous rigid motion (a path in \(\mathrm{SE}(2)\)) that moves \(S\) from the horizontal arm to the vertical arm while keeping \(S\subset\mathcal C\) at every instant.

Known Bounds

Quantity Value Year Author
Lower bound \(A_{\text{Gerver}} \approx 2.21953166887197\) 1992 Joseph Gerver
Upper bound \(A_{\text{upper}} \le 2.37\) 2018 Yoav Kallus

Gerver’s shape is piecewise-smooth, composed of 18 distinct analytic curves, and is conjectured to be optimal.

Key Mathematical Tools

  • Minkowski sums for swept-area calculations

  • Support functions to describe convex bodies

  • Variational calculus to derive necessary optimality conditions

  • Hamilton–Jacobi–Bellman equations for motion-planning constraints

Open Questions

  1. Exact value: Is Gerver’s sofa truly area-maximal?

  2. Uniqueness: Is the maximizer unique up to rigid motion?

  3. Higher dimensions: What is the largest 3-D sofa that can round two perpendicular unit-width corridors?

  4. Computational complexity: Is there a polynomial-time algorithm to approximate \(A_{\max}\) within \(\varepsilon\)?

References

  • Gerver, J. L. (1992). On moving a sofa around a corner. Geometriae Dedicata, 42(3), 267–283.

  • Kallus, Y. (2018). A new upper bound on the moving sofa problem. arXiv:1706.06630.

  • Romik, D. (2016). The Moving Sofa Problem. Newsletter of the European Mathematical Society, 101, 31–35.