Convex metric

An internal metric on a two-dimensional manifold  which meets some convexity condition. More exactly, let  and  be two shortest lines issuing from some point ; let  and  be points on these lines; let  be the distances from  to  and , respectively; let  be the distance between  and ; and let  be the angle opposite to the side  in the plane triangle with sides . The convexity condition of the metric (at the point ) is that  is a non-increasing function (i.e.  if , ) on any pair of intervals ,  such that the points  and , which correspond to two arbitrary values in these intervals, can be connected by a shortest line. An internal metric is a convex metric if and only if it is a metric of non-negative curvature. The metric of a convex surface is a convex metric. Conversely, any two-dimensional manifold with a convex metric can be realized as a convex surface (Aleksandrov's theorem).

#### References[edit]

[1] |  A.D. Aleksandrov, "Die innere Geometrie der konvexen Flächen" , Akademie Verlag (1955) (Translated from Russian)

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