Moving sofa problem

In November 2024, Jineon Baek posted an arXiv preprint claiming that Gerver's value is optimal, which if true, would solve the moving sofa problem.

[2] The first formal publication was by the Austrian-Canadian mathematician Leo Moser in 1966,[3] although there had been many informal mentions before that date.

A lower bound on the sofa constant can be proven by finding a specific shape of a high area and a path for moving it through the corner.

[4] This can be achieved using a shape resembling an old-fashioned telephone handset, consisting of two quarter-disks of radius 1 on either side of a 1 by

[5][6] In 1992, Joseph L. Gerver of Rutgers University described a sofa with 18 curve sections, each taking a smooth analytic form.

This further increased the lower bound for the sofa constant to approximately 2.2195 (sequence A128463 in the OEIS).

[4][1][9] Yoav Kallus and Dan Romik published a new upper bound in 2018, capping the sofa constant at

Their approach involves rotating the corridor (rather than the sofa) through a finite sequence of distinct angles (rather than continuously) and using a computer search to find translations for each rotated copy so that the intersection of all of the copies has a connected component with as large an area as possible.

As they show, this provides a valid upper bound for the optimal sofa, which can be made more accurate using more rotation angles.

Five carefully chosen rotation angles lead to the stated upper bound.