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  "body": "\n## Problem\n\nLet $U$ be an open convex subset of $\\R^n$.\nDetermine all [[metrics|0VKD1TP7F3]] $d$ on $U$ for which every straight [[line|0MP9BCF6DP]] segment in $U$ is a shortest path.\nThat is, determine all $d$ such that $d(x, z) = d(x, y) + d(y, z)$ whenever $y$ lies on the segment from $x$ to $z$.\n\n## Remarks\n\nSuch metrics are called projective metrics, and their [[geodesics|NSYMQMHPDN]] include the straight lines.\nExamples are the Euclidean metric and the metrics given by any norm on $\\R^n$.\nAnother example is the Hilbert metric on a bounded convex domain; when the domain is a disc, it gives the Beltrami–Klein model of the hyperbolic plane.\nHilbert posed the question in his list of problems of 1900.\nHerbert Busemann gave a construction of such metrics from measures on the space of hyperplanes.\nAleksei Pogorelov proved that in the plane every continuous projective metric arises in this way.\n\n## Explanations\n\n### L1 human\n\nIn ordinary geometry the shortest way between two points is the straight line.\nHilbert asked which other ways of measuring distance keep this property.\nThere are many: for example, the taxicab distance $\\abs{x_1 - y_1} + \\abs{x_2 - y_2}$ in the plane also makes every straight segment a shortest path, although it is not the only shortest path.\n",
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