Improving Upper Bound for the Moving Sofa Problem Based on Quadratic Programming

The moving sofa problem is a problem in discrete geometry that involves finding a shape with the maximum area that can pass through a right-angled corner in a hallway with unit width, and has remained an open problem since it was posed by Moser in 1966. The largest lower bound on the maximum area known to date is 2.2195..., found by Gerver in 1992, and the smallest upper bound is 2.37, proved by Kallus and Romik in 2018. While the minimum upper bound of 2.37 appears to be somewhat loose, experimental evidence suggests that there is little room for improvement in the largest lower bound of 2.2195.... In this study, we aim to lower the upper bound to be closer to Gerver’s maximum lower bound. For this purpose, we construct an optimization algorithm based on geometric branch-and-bound and quadratic programming, and prove various geometric properties for efficient search of the algorithm. Under a reasonable geometric assumption, the proposed optimization algorithm obtains an improved upper bound of 2.3361 with much shorter computation time than Kallus and Romik.