The Ford-Circle Packing Has Maximum Area: An Answer to a Question of Propp and Kenyon

Two disks of radius 1 centred at (±1, 1) and the x-axis enclose a curvilinear triangle. In the open problem session of the 2015 Oberwolfach workshop on discrete differential geometry, Propp and Kenyon asked whether, among all packings of this triangle by disks that touch the x-axis, the greedy packing has the largest total area. The greedy packing places each new disk in an interstice so that it touches the two disks bounding the interstice. It is a rescaled copy of the Ford circles, and its area is π(ζ(3)/ζ(4) − 1) ≈ 0.34754. We prove that the answer is yes. More generally, let A and B be tangent disks resting on a line, and let G be the greedy packing of the gap between them. For every packing P of this gap by disks resting on the line, and for every α > 1, we show that Σ_{D∈P} r_D^α ≤ Σ_{D∈G} r_D^α, where r_D is the radius of D. The main step is the same inequality for the weight 1/(e^{1/√r} − 1), for which the greedy value of the gap is the product of the weights of A and B. Powers of the radius are superpositions of rescaled copies of this weight. An optimal finite packing contains a chain of tangent disks from A to B, and we bound the value of such a chain by moving two consecutive disks at a time. Along such a move the value is, up to an additive constant, a product of two log-convex functions. The log-convexity reduces to the positivity of an explicit function of two variables whose double power series has non-negative coefficients. We do not discuss uniqueness of the maximizer. This is an unrefereed note.