Editing Hilbert's fourth problem (section)

==Three dimensional case==

For three dimensional case Pogorelov proved the following theorem.

'''Theorem.''' ''Any three-dimensional regular complete flat metric is a <math>\sigma</math>-metric.''

However, in the three-dimensional case <math>\sigma</math>-measures can take either positive or negative values. The  necessary and sufficient conditions for the regular metric defined by the  function of the set <math>\sigma</math> to be flat are the following three conditions:
# the value <math>\sigma</math> on any plane equals zero,
# the value <math>\sigma</math> in any cone is non-negative,
# the value <math>\sigma</math> is positive if the cone contains interior points.

Moreover, Pogorelov showed that any complete continuous flat metric in the three-dimensional case is the limit of regular <math>\sigma</math>-metrics with the uniform convergence on any compact sub-domain of the metric's domain. He called them  generalized <math>\sigma</math>-metrics.

Thus Pogorelov could prove the following statement.

'''Theorem.''' ''In the three-dimensional case any complete continuous flat metric is a <math>\sigma</math>-metric in generalized meaning.''

Busemann, in his review to Pogorelov’s book "Hilbert’s Fourth Problem" wrote, "In the spirit of the time Hilbert restricted himself to ''n'' = 2, 3 and so does Pogorelov.
However, this has doubtless pedagogical reasons, because he addresses a wide class of readers. The real difference is between ''n'' = 2 and ''n''>2. Pogorelov's method works for ''n''>3, but requires greater technicalities".<ref>{{cite journal
| last1=Busemann | first1=Herbert | authorlink1=Herbert Busemann
| title=Review of: A. V. Pogorelov, Hilbert's fourth problem
| journal=Bulletin of the American Mathematical Society
| series=New Series
| volume=4
| issue=1
| date=1981
| pages=87–90
| doi=10.1090/S0273-0979-1981-14867-9 | url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-4/issue-1/Review-A-V-Pogorelov-Hilberts-fourth-problem/bams/1183547850.full| doi-access=free
}}</ref>