Moving sofa problem

The Hammersley sofa has area 2.2074 but is not the largest solution

Gerver's sofa of area 2.2195 with 18 curve sections

Lower bounds can be proven by finding a specific shape of high area and a path for moving it through the corner. An obvious lower bound is ${\displaystyle A\geq \pi /2\approx 1.57}$. This comes from a sofa that is a half-disk of unit radius, which can slide up one passage into the corner, rotate within the corner around the center of the disk, and then slide out the other passage.

In 1968, John Hammersley stated a lower bound of ${\displaystyle A\geq \pi /2+2/\pi \approx 2.2074}$.[3] This can be achieved using a shape resembling a telephone handset, consisting of two quarter-disks of radius 1 on either side of a 1 by ${\displaystyle 4/\pi }$ rectangle from which a half-disk of radius ${\displaystyle 2/\pi }$ has been removed.[4][5]

In 1992, Joseph L. Gerver of Rutgers University described a sofa specified by 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).[6][7]

Hammersley stated an upper bound on the sofa constant of at most ${\displaystyle 2{\sqrt {2}}\approx 2.8284}$.[3][1][8] Yoav Kallus and Dan Romik published a new upper bound in 2018, capping the sofa constant at ${\displaystyle 2.37}$. 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 by using more rotation angles. A set of five carefully-chosen rotation angles leads to the stated upper bound.[9]