Short simple orthogonal geodesic chords on a 2-disk with convex boundary

We prove the existence of at least two distinct short, simple orthogonal geodesic chords on a Riemannian 2-disk [Formula: see text] with convex boundary. The lengths of these curves are bounded in terms of the length of [Formula: see text], the diameter of [Formula: see text], and the area of [Formula: see text]. We also prove the existence of a short, simple geodesic chord on any Riemannian 2-disk (with or without convex boundary) that is orthogonal to the disk’s boundary at least one endpoint.