Existence and Non-existence of Half-Geodesics on S2S^2

In this paper we study half-geodesics, those closed geodesics that minimize on any subinterval of length $l(\gamma)/2$. For each nonnegative integer $n$, we construct Riemannian manifolds diffeomorphic to $S^2$ admitting exactly $n$ half-geodesics. Additionally, we construct a sequence of Riemannian manifolds, each of which is diffeomorphic to $S^2$ and admits no half-geodesics, yet which converge in the Gromov-Hausdorff sense to a limit space with infinitely many half-geodesics.Comment: This version has been shortened for publicatio

Characterizing Round Spheres Using Half-Geodesics

A half-geodesic is a closed geodesic realizing the distance between any pair of its points. All geodesics in a round sphere are half-geodesics. Conversely, this note establishes that Riemannian spheres with all geodesics closed and sufficiently many half-geodesics are round

An Economic Model of Fair Use

The doctrine of fair use allows limited copying of creative works based on the rationale that copyright holders would consent to such uses if bargaining were possible. This paper develops a formal model of fair use in an effort to derive the efficient legal standard for applying the doctrine. The model interprets copies and originals as differentiated products and defines fair use as a threshold separating permissible copying from infringement. The analysis highlights the role of technology in shaping the efficient standard. Discussion of several key cases illustrates the applicability of the model.Fair use, Copyright law, Technological improvement