Hilbert's fourth problem

In one statement derived from the original, it was to find — up to an isomorphism — all geometries that have an axiomatic system of the classical geometry (Euclidean, hyperbolic and elliptic), with those axioms of congruence that involve the concept of the angle dropped, and `triangle inequality', regarded as an axiom, added.

[3][4] In 1976, Armenian mathematician Rouben V. Ambartzumian proposed another proof of Hilbert's fourth problem.

[6] Due to the idea that a 'straight line' is defined as the shortest path between two points, he mentions how congruence of triangles is necessary for Euclid's proof that a straight line in the plane is the shortest distance between two points.

He summarizes as follows: The theorem of the straight line as the shortest distance between two points and the essentially equivalent theorem of Euclid about the sides of a triangle, play an important part not only in number theory but also in the theory of surfaces and in the calculus of variations.

The necessary condition for solving Hilbert's fourth problem is the requirement that a metric space that satisfies the axioms of this problem should be Desarguesian, i.e.,: For Desarguesian spaces Georg Hamel proved that every solution of Hilbert's fourth problem can be represented in a real projective space

if one determines the congruence of segments by equality of their lengths in a special metric for which the lines of the projective space are geodesics.

Hamel solved this problem under the assumption of high regularity of the metric.

The axioms of geometries under consideration imply only a continuity of the metrics.

Therefore, to solve Hilbert's fourth problem completely it is necessary to determine constructively all the continuous flat metrics.

Before 1900, there was known the Cayley–Klein model of Lobachevsky geometry in the unit disk, according to which geodesic lines are chords of the disk and the distance between points is defined as a logarithm of the cross-ratio of a quadruple.

be a compact convex hypersurface in a Euclidean space defined by where the function

be a bounded open convex set with the boundary of class C2 and positive normal curvatures.

It is defined in a domain bounded by a closed convex hypersurface and is also flat.

Georg Hamel was first to contribute to the solution of Hilbert's fourth problem.

is flat if and only if it satisfies the conditions: Consider a set of all oriented lines on a plane.

Then the set of all oriented lines is homeomorphic to a circular cylinder of radius 1 with the area element

In 1966, in his talk at the International Mathematical Congress in Moscow, Herbert Busemann introduced a new class of flat metrics.

Thus Hilbert's fourth problem for the two-dimensional case was completely solved.

A consequence of this is that you can glue boundary to boundary two copies of the same planar convex shape, with an angle twist between them, you will get a 3D object without crease lines, the two faces being developable.

However, this structure can not be generalized to higher dimensions because of Hilbert's third problem solved by Max Dehn.

The necessary and sufficient conditions for the regular metric defined by the function of the set

However, this has doubtless pedagogical reasons, because he addresses a wide class of readers.

[12] The multi-dimensional case of the Fourth Hilbert problem was studied by Szabo.

Szabo described all continuous flat metrics in terms of generalized functions.

Hilbert's fourth problem is also closely related to the properties of convex bodies.

A convex polyhedron is called a zonotope if it is the Minkowski sum of segments.

A convex body which is a limit of zonotopes in the Blaschke – Hausdorff metric is called a zonoid.

The Minkowski space is generated by the Blaschke–Busemann construction if and only if the support function of the indicatrix has the form of (1), where

From the above statement it follows that the flat metric of Minkowski space with the norm

There was found the correspondence between the planar n-dimensional Finsler metrics and special symplectic forms on the Grassmann manifold