A description of linearly additive metrics on $ \mathbb{R}^n$

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.