Riemannian distance and symplectic embeddings in cotangent bundle

Given an open neighborhood Formula: see text of the zero section in the cotangent bundle of Formula: see text we define a distance-like function Formula: see text on Formula: see text using certain symplectic embeddings from the standard ball Formula: see text to Formula: see text. We show that when Formula: see text is the unit-disk cotangent bundle of a Riemannian metric on Formula: see text, Formula: see text recovers the metric. As an intermediate step, we give a new construction of a symplectic embedding of the ball of capacity 4 to the product of Lagrangian disks Formula: see text, and we give a new proof of the strong Viterbo conjecture about normalized capacities for Formula: see text. We also give bounds of the symplectic packing number of two balls in a unit-disk cotangent bundle relative to the zero section Formula: see text.