Editing Hilbert's fourth problem (section)

=== Ambartsumian's proofs===
In 1976, Ambartsumian proposed another proof of Hilbert's fourth problem.<ref name="Ambartzumian1976" />

His proof uses the fact  that in the two-dimensional case  the whole measure can be restored by its values on biangles, and thus be  defined on triangles in the same way as the area of a triangle is defined on a sphere. Since the triangle inequality holds, it follows that this measure is positive on non-degenerate triangles and  is determined on all [[Borel set]]s. However, this structure can not be generalized to higher dimensions because of Hilbert's third problem solved by [[Max Dehn]].

In the two-dimensional case,  polygons with the same volume are scissors-congruent. As was shown by Dehn this is not true for a higher dimension.