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].