A Conditional Upper Bound for the Moving Sofa Problem

The moving sofa problem asks for the connected shape with the largest area ₌₀ₗ that can move around the right-angled corner of a hallway L with unit width. The best bounds currently known on _ are summarized as 2. 2195 _ 2. 37. The lower bound 2. 2195 _ comes from Gerver's sofa SG of area G: = 2. 2195. The upper bound _ 2. 37 was proved by Kallus and Romik using extensive computer assistance. It is conjectured that the equality _ = 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 + ²/8 = 2. 2337. The new conditional bound does not rely on any computer assistance, yet it is much closer to the lower bound 2. 2195 of Gerver than the computer-assisted upper bound 2. 37 of Kallus and Romik. Gerver's sofa SG, the conjectured optimum, satisfies the injectivity condition in particular.