Editing Hilbert's fourth problem (section)

==''σ''-metrics==

===Sufficient condition for flat metrics===
[[Georg Hamel]] was first to contribute to the solution of Hilbert's fourth problem.<ref name="Hamel1903"/> He proved the following statement.

'''Theorem'''. A regular Finsler metric <math>F(x,y)=F(x_1,\ldots,x_n,y_1,\ldots,y_n)</math> is flat if and only if it satisfies the conditions:

: <math>\frac{\partial^2 F^2}{\partial x^i \, \partial y^j} = \frac{\partial^2 F^2}{\partial x^j \, \partial y^i}, \, i,j=1,\ldots,n.</math>

===Crofton formula===
{{main|Crofton formula}}
Consider a set of all oriented lines on a plane. Each line is defined by the parameters <math>\rho</math> and <math>\varphi,</math> where <math>\rho</math> is a distance from the origin  to the  line, and <math>\varphi</math> is an angle between the  line and the ''x''-axis. Then the set of all oriented lines is homeomorphic to a circular cylinder of radius 1 with the area element <math>dS = d\rho \, d\varphi </math>. Let <math>\gamma</math> be a rectifiable curve on a plane. Then the length of <math>\gamma</math> is
<math display="block">L = \frac{1}{4} \iint_\Omega n( \rho, \varphi) \, dp \, d\varphi</math>
where <math>\Omega</math> is a set of lines that intersect the curve <math>\gamma</math>, and <math>n(p, \varphi)</math> is the number of intersections of the line with <math>\gamma</math>.
Crofton proved this statement in 1870.<ref>{{cite book
| last1=Santaló | first1=Luís A. | authorlink1=Luis Santaló
| chapter=Integral geometry
| title=Studies in Global Geometry and Analysis
| editor-last1=Chern | editor-first1=S. S.
| publisher=Mathematical Association of America, Washington, D. C.
| pages=147–195
| date=1967}}</ref>

A similar statement holds for a projective space.

=== Blaschke–Busemann  measure ===
In 1966, in his talk at the [[International Congress of Mathematicians|International Mathematical Congress]] in Moscow, [[Herbert Busemann]] introduced a new class of flat metrics. On a set of lines on the projective plane <math>RP^{2}</math> he introduced a completely additive non-negative measure <math>\sigma</math>, which satisfies the following conditions:

# <math>\sigma (\tau P)=0</math>, where <math>\tau P</math> is a set of straight lines  passing through a point ''P'';
# <math>\sigma (\tau X)>0</math>, where <math>\tau X</math> is a set of straight lines passing through some set ''X'' that contains a straight line segment;
# <math>\sigma (RP^{n})</math> is finite.

If we consider a <math>\sigma</math>-metric in an arbitrary convex domain <math>\Omega</math> of a projective space <math>RP^{2}</math>, then condition  3) should be  replaced by the following:
for any set ''H'' such that ''H'' is contained in <math>\Omega</math> and the closure of ''H'' does not intersect the boundary of <math>\Omega</math>, the inequality
: <math>\sigma(\pi H)<\infty</math> holds.<ref name="Buseman1955">{{cite book
| last1=Busemann | first1=Herbert | authorlink1=Herbert Busemann
| title=The Geometry of Geodesics
| publisher=Academic Press, New York
| date=1955}}</ref>

Using this measure,  the <math>\sigma</math>-metric  on <math>RP^{2}</math> is defined by
: <math>|x,y|=\sigma \left(  \tau [x,y]  \right),</math>
where <math>\tau [x,y]</math> is the set of straight lines that intersect the segment <math>[x,y]</math>.

The triangle inequality for this metric follows from [[Pasch's theorem]].

'''Theorem'''. <math>\sigma</math>-metric on <math>RP^{2}</math> is flat, i.e., the geodesics are the straight lines of the projective space.

But Busemann was far from the idea that <math>\sigma</math>-metrics exhaust all flat metrics. He wrote, ''"The freedom in the choice of a metric with given geodesics is for non-Riemannian metrics so great that it may be doubted whether there really exists a convincing characterization of all Desarguesian spaces"''.<ref name="Buseman1955"/>