Solving Moving Sofa Problem Using Calculus of Variations

TL;DR

This paper applies calculus of variations to analytically approach the longstanding moving sofa problem, deriving a shape with a specific maximal area and confirming consistency with known solutions.

Contribution

It introduces a calculus of variations framework to solve the moving sofa problem, providing a new analytical method and numerical results for the maximal sofa area.

Findings

Derived a shape with area 2.2195316 using calculus of variations.

Confirmed that Gerver's sofa and Romik's car satisfy the Euler-Lagrange equations.

Explored asymmetric cases and variant problems related to the moving sofa problem.

Abstract

In 1966, Leo Moser introduced the "moving sofa problem," which seeks to determine the largest area of a shape that can be maneuvered through a 90-degree hallway of unit-width. This problem remains unsolved and open yet. In this paper, we employ calculus of variations method to solve this problem. Assuming the trajectories and envelopes are convex, the sofa's area is formulated as an integral functional on a set of parametric equations for curves. The final shape is determined by solving the Euler-Lagrange equations. Utilizing numerical methods, we obtain the non-trivial area of 2.2195316, consistent with the previously well-known Gerver's constant since 1992. We prove that both the results of Gerver's sofa and Romik's car satisfy the Euler-Lagrange equations for the necessary condition of maximal area. We also explore additional cases and asymmetric conditions, and discuss other variant problems.