Geodesic
A description of linearly additive metrics on $ \mathbb{R}^n$
Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 81 (2020) no. 1, pp. 137-144

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There is an integral-geometric approach, proposed by Busemann, for building linearly additive metrics on $ \mathbb{R}^n $ (it uses hyperplanes). Hilbert's Fourth Problem was solved with the help of this construction. In this article, we present a new description (using straight lines) of linearly additive metrics on $ \mathbb{R}^n$, generated by a norm. There is a link between this description and the sine transform. 
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     author = {R. H. Aramyan},
     title = {A description of linearly additive metrics on $ \mathbb{R}^n$},
     journal = {Trudy Moskovskogo matemati\v{c}eskogo ob\^{s}estva},
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R. H. Aramyan. A description of linearly additive metrics on $ \mathbb{R}^n$. Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 81 (2020) no. 1, pp. 137-144. http://geodesic.mathdoc.fr/item/MMO_2020_81_1_a5/