Editing Hilbert's fourth problem (section)

==Multidimensional case==
The multi-dimensional case of the Fourth Hilbert problem was studied by Szabo.<ref>{{cite journal
| last1=Szabó | first1=Z. I.
| title=Hilbert's fourth problem I
| journal=[[Advances in Mathematics]]
| volume=59
| issue=3
| date=1986
| pages=185–301
| doi=10.1016/0001-8708(86)90056-3 | doi-access=free}}</ref> In 1986, he proved, as he wrote, the generalized Pogorelov theorem.

'''Theorem.''' Each ''n''-dimensional Desarguesian  space  of the class <math>C^{n+2}, n>2</math>, is generated by the Blaschke–Busemann construction.

A <math>\sigma</math>-measure that generates a flat measure has the following properties:
# the <math>\sigma</math>-measure of hyperplanes passing through a fixed point is equal to zero;
# the <math>\sigma</math>-measure of the set of hyperplanes  intersecting two segments [''x'', ''y''], [''y'', ''z''], where ''x'', ''y'' та ''z'' are not collinear, is positive.

There was given the example of a flat metric not generated by the Blaschke–Busemann construction. Szabo described all continuous flat metrics in terms of generalized functions.