Convex metric

An internal metric on a two-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026280/c0262801.png" /> which meets some convexity condition. More exactly, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026280/c0262802.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026280/c0262803.png" /> be two shortest lines issuing from some point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026280/c0262804.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026280/c0262805.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026280/c0262806.png" /> be points on these lines; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026280/c0262807.png" /> be the distances from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026280/c0262808.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026280/c0262809.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026280/c02628010.png" />, respectively; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026280/c02628011.png" /> be the distance between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026280/c02628012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026280/c02628013.png" />; and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026280/c02628014.png" /> be the angle opposite to the side <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026280/c02628015.png" /> in the plane triangle with sides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026280/c02628016.png" />. The convexity condition of the metric (at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026280/c02628017.png" />) is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026280/c02628018.png" /> is a non-increasing function (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026280/c02628019.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026280/c02628020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026280/c02628021.png" />) on any pair of intervals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026280/c02628022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026280/c02628023.png" /> such that the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026280/c02628024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026280/c02628025.png" />, 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

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