Editing Hilbert's fourth problem (section)

=== Hilbert's geometry ===
[[Image:Gilbert metric.svg|thumb|Hilbert's metric]]
Let <math>U\subset (\mathbb{E}^{n+1}, \| \cdot \|_{\mathbb{E}})</math> be a bounded open convex set with  the boundary of  class ''C<sup>2</sup>'' and positive normal curvatures. Similarly to the Lobachevsky space, the hypersurface <math>\partial U</math> is called the absolute of Hilbert's geometry.<ref>{{cite journal
| last1=Hilbert | first1=David | authorlink1=David Hilbert
| title=Uber die gerade Linie als kürzeste Verbindung zweier Punkte
| journal=Mathematische Annalen
| volume=46
| date=1895
| pages=91–96
| doi=10.1007/BF02096204 | doi-access=free}}</ref>

Hilbert's distance (see fig.) is  defined by
: <math>d_U(p, q)=\frac{1}{2} \ln \frac{\|q-q_1\|_E}{\|q-p_1\|_E}\times \frac{\|p-p_1\|_E}{\|p-q_1\|_E}.</math>

[[Image:Finsler metric.svg|thumb|Hilbert–Finsler metric]]
The distance <math>d_{U}</math> induces the '''Hilbert–Finsler metric''' <math>F_{U}</math> on ''U''. For any  <math>x\in U</math> and <math>y\in T_{x}U</math> (see fig.), we have
: <math>F_U(x, y)=\frac{1}{2}\|y\|_{\mathbb{E}} \left(    \frac{1}{\|x-x_{+}\|_{\mathbb{E}}}+\frac{1}{\|x-x_{-}\|_{\mathbb{E}}}   \right). </math>

The metric is symmetric and flat. In 1895, Hilbert introduced this metric as a generalization of  the  Lobachevsky geometry. If the hypersurface <math>\partial U </math> is an ellipsoid, then we have the Lobachevsky geometry.