DELAUNAY TRIANGULATIONS OF EXTRA - LARGE METRICS 3 Proof

In this note we study extra large spherical conemanifolds in dimension 2 (though many of our resultsz and techniques extend to higher dimensions. A 2-dimensional spherical cone manifold is a metric space where all but finitely many points has a neighborhood isometric to a neighborhood of a point on the round sphere S. The exceptional points (cone points) have neighboroods isometric to a spherical cone, the angle of which is the cone angle at that point. If M is a cone manifold, we define a geodesic to be a locally length minimizing curve onM. It is easy to see that such a curve is locally a great circle, except at the cone points. There, the geodesic must have the property that it subtends an angle no smaller than π on either side. Consequently, no geodesics can pass through cone points, where the cone angles are smaller than 2π (such cone points are known as positively curved cone points, since the curvature of a cone point is defined as 2π less the cone angle at the point). We say that a spherical cone manifold is extra large if