A Conditional Upper Bound for the Moving Sofa Problem

Abstract

The moving sofa problem asks for the connected shape with the largest area $\mu_{\text{max}}$ that can move around the right-angled corner of a hallway $L$ with unit width. The best bounds currently known on $\mu_{\max}$ are summarized as $2.2195\ldots \leq \mu_{\max} \leq 2.37$. The lower bound $2.2195\ldots \leq \mu_{\max}$ comes from Gerver's sofa $S_G$ of area $\mu_G := 2.2195\ldots$. The upper bound $\mu_{\max} \leq 2.37$ was proved by Kallus and Romik using extensive computer assistance. It is conjectured that the equality $\mu_{\max} = \mu_G$ holds at the lower bound. We develop a new approach to the moving sofa problem by approximating it as an infinite-dimensional convex quadratic optimization problem. The problem is then explicitly solved using a calculus of variation based on the Brunn-Minkowski theory. Consequently, we prove that any moving sofa satisfying a property named the injectivity condition has an area of at most $1 + \pi^2/8 = 2.2337\dots$. The new conditional bound does not rely on any computer assistance, yet it is much closer to the lower bound $2.2195\ldots$ of Gerver than the computer-assisted upper bound $2.37$ of Kallus and Romik. Gerver's sofa $S_G$, the conjectured optimum, satisfies the injectivity condition in particular.