A Numerical Approach for Analysing the Moving Sofa Problem

Batsch, Michał

doi:10.3390/sym14071409

Symmetry

2022

DOI: 10.3390/sym14071409

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A Numerical Approach for Analysing the Moving Sofa Problem

Michał Batsch¹

Abstract: This paper presents a method for obtaining the shape and area of a sofa. The proposed approach is based on a discrete solution to the equation, which states the necessary conditions for the existence of envelopes. Based on provided examples, it was proved that the method can be used for deriving the solutions of the posed problem. The method offers an area calculation accuracy of 1.5×10−8.

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2024

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Deep Learning Evidence for Global Optimality of Gerver’s Sofa

Leng,

Bi,

Cha

et al. 2024

Symmetry

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The moving sofa problem, introduced by Leo Moser in 1966, seeks to determine the maximal area of a 2D shape that can navigate an L-shaped corridor of unit width. Joseph Gerver’s 1992 solution, providing a lower bound of approximately 2.2195, is the best known, though its global optimality remains unproven. This paper leverages neural networks’ approximation power and recent advances in invexity optimization to explore global optimality. We propose two approaches supporting Gerver’s conjecture that his sofa is the unique global maximum. The first approach uses continuous function learning, discarding assumptions about the monotonicity, symmetry, and differentiability of sofa movements. The sofa area is computed as a differentiable function using our “waterfall” algorithm, with the loss function incorporating both differential terms and initial conditions based on physics-informed machine learning. Extensive training with diverse network initialization consistently converges to Gerver’s solution. The second approach applies discrete optimization to the Kallus–Romik upper bound, improving it from 2.37 to 2.3337 for five rotation angles. As the number of angles increases, our model asymptotically converges to Gerver’s area from above, indicating that no larger sofa exists.

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Deep Learning Evidence for Global Optimality of Gerver’s Sofa

Leng,

Bi,

Cha

et al. 2024

Symmetry

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A Numerical Approach for Analysing the Moving Sofa Problem

Cited by 1 publication

References 14 publications

Deep Learning Evidence for Global Optimality of Gerver’s Sofa

Deep Learning Evidence for Global Optimality of Gerver’s Sofa

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