The fourth problem of Hilbert

Hilbert's fourth problem in the list of problems of Hilbert concerns the basic axiom system of geometry . The problem is that

“Determine everything up to the isomorphism of the realization of the axioms systems of classical geometries (Euclidean, Lobachevsky and elliptic) if we omit the congruence axioms containing the notion of angle in them and supplement these systems with the triangle axiom of inequality” ^[1] .

In the case of a plane, if we take the continuity axiom, we arrive at the problem posed by Darboux:

“Find on the plane all variational problems whose solutions are all straight lines on the plane” ^[2] .

Flat metrics

The straight lines connecting the corresponding vertices of the triangles pass through one point - it is located on the left. Accordingly, the three points at which the continuations of the three pairs of the corresponding sides of the triangles intersect lie on one straight line — it is purple.

The Desargue theorem is true :
If two triangles are located on a plane in such a way that the straight lines connecting the corresponding vertices of the triangles pass through one point, then the three points at which the extensions of three pairs of corresponding sides of the triangles intersect lie on one straight line

A prerequisite for solving the IV problem of Hilbert is the requirement that the metric space satisfying the axioms of this problem be desargistic, that is, the following conditions must be met:

If the space is two-dimensional, it is necessary that the Desargue theorem and the inverse to it be fulfilled;
If the dimension of the space is more than two, then it is necessary that any three points lie on the same plane.

For desarg spaces proved that every solution of the Hilbert problem can be represented in a real projective space $RP^{n}$ ${\ displaystyle RP ^ {n}}$ or in a convex area $RP^{n},$ ${\ displaystyle RP ^ {n},}$ if we determine the congruence of segments by the equality of their lengths in a special metric for which the lines of the projective space are geodesic.

Such metrics are called flat or projective.

Thus, the solution of the Hilbert problem was reduced to the problem of constructively determining all complete flat metrics.

Hamel solved this problem by proposing sufficient metric regularity ^[3] . However, as simple examples show, regular flat metrics are far from exhausting all flat metrics. From the axioms of the considered geometry, only the continuity of metrics follows. Therefore, a complete solution to the Hilbert problem implies a constructive definition of all continuous flat metrics.

Prehistory IV of the Hilbert Problem

Cayley-Klein model of Lobachevsky geometry

Until 1900, Cayley-Klein's interpretation of Lobachevsky’s geometry in the unit circle was known, where the straight lines are the chords of the circle, and the distance between points was defined as the logarithm of the complex ratio of four points.

For two-dimensional Riemannian metrics, E. Beltrami (1835–1900) proved that the only flat metrics are metrics of constant curvature ^[4] .

For multidimensional Riemannian metrics, this statement was proved by E. Cartan in 1930.

In 1890, in connection with number theory, G. Minkowski introduced what we now call finite-dimensional Banach spaces ^[5] .

Minkowski space

$F_{0}\subset \mathbb {E} ^{n}$ ${\ displaystyle F_ {0} \ subset \ mathbb {E} ^ {n}}$ - compact closed convex hypersurface in Euclidean space, given implicitly

F_{0}=\{y\in E^{n}:F(y)=1\}.

{\ displaystyle F_ {0} = \ {y \ in E ^ {n}: F (y) = 1 \}.}

Function $F=F(y)$ ${\ displaystyle F = F (y)}$ satisfies the conditions:

$F(y)\geqslant 0,\qquad F(y)=0\Leftrightarrow y=0$ ${\ displaystyle F (y) \ geqslant 0, \ qquad F (y) = 0 \ Leftrightarrow y = 0}$ ;
$F(\lambda y)=\lambda F(y),\qquad \lambda \geqslant 0$ ${\ displaystyle F (\ lambda y) = \ lambda F (y), \ qquad \ lambda \ geqslant 0}$ ;
$F(y)\in C^{k}(E^{n}\setminus \{0\}),\qquad k\geqslant 3$ {\ displaystyle F (y) \ in C ^ {k} (E ^ {n} \ setminus \ {0 \}), \ qquad k \ geqslant 3} ;
${\frac {\partial ^{2}F^{2}}{\partial y^{i}\partial y^{j}}}\xi ^{i}\xi ^{j}>0$ ${\ displaystyle {\ frac {\ partial ^ {2} F ^ {2}} {\ partial y ^ {i} \ partial y ^ {j}}} \ xi ^ {i} \ xi ^ {j}> 0 }$ .

Set the length of the vector OA as:

||OA||_{M}={\frac {||OA||_{\mathbb {E} }}{||OL||_{\mathbb {E} }}}.

{\ displaystyle || OA || _ {M} = {\ frac {|| OA || _ {\ mathbb {E}}} {|| OL || _ {\ mathbb {E}}}}.}

A space with such a metric is called a Minkowski space.

Hypersurface $F_{0}$ ${\ displaystyle F_ {0}}$ may be irregular convex surface. The metric defined in this way is flat.

Finsler spaces

Let M be a smooth, finite-dimensional manifold, $(x,y)\in TM$ ${\ displaystyle (x, y) \ in TM}$ - tangent bundle. Function $F(x,y)\colon TM\rightarrow [0,+\infty )$ ${\ displaystyle F (x, y) \ colon TM \ rightarrow [0, + \ infty)}$ called Finsler metric if

$F(x,y)\in C^{k}(TM\setminus \{0\}),\qquad k\geqslant 3$ ${\ displaystyle F (x, y) \ in C ^ {k} (TM \ setminus \ {0 \}), \ qquad k \ geqslant 3}$ ;
For each point $x\in M$ ${\ displaystyle x \ in M}$ function restriction $F(x,y)$ ${\ displaystyle F (x, y)}$ on $T_{x}M$ ${\ displaystyle T_ {x} M}$ is the Minkowski norm.

$(M,F)$ ${\ displaystyle (M, F)}$ called finsler space.

Hilbert's geometry

Hilbert metric

$U\subset (\mathbb {E} ^{n+1},\|.\|_{\mathbb {E} })$ ${\ displaystyle U \ subset (\ mathbb {E} ^ {n + 1}, \ |. \ | _ {\ mathbb {E}})}$ - bounded open convex set with boundary of class C ² and positive normal curvatures. By analogy with the Lobachevsky space, a hypersurface $\partial U$ ${\ displaystyle \ partial U}$ is called the absolute of Hilbert's geometry ^[6] .

Hilbert metric

d_{U}(p,q)={\frac {1}{2}}\ln {\frac {\|q-q_{1}\|_{E}}{\|q-p_{1}\|_{E}}}\times {\frac {\|p-p_{1}\|_{E}}{\|p-q_{1}\|_{E}}}.

{\ displaystyle d_ {U} (p, q) = {\ frac {1} {2}} \ ln {\ frac {\ | q-q_ {1} \ | _ {E}} {\ | q-p_ {1} \ | _ {E}}} \ times {\ frac {\ | p-p_ {1} \ | _ {E}} {\ | p-q_ {1} \ | _ {E}}}. }

Finsler's metric of Hilbert

$d_{U}$ ${\ displaystyle d_ {U}}$ induces Hilbert's Finsler metric $F_{U}$ ${\ displaystyle F_ {U}}$ on u for any $x\in U$ ${\ displaystyle x \ in U}$ and $y\in T_{x}U$ ${\ displaystyle y \ in T_ {x} U}$ (see fig.)

F_{U}(x,y)={\frac {1}{2}}\|y\|_{\mathbb {E} }\left({\frac {1}{\|x-x_{+}\|_{\mathbb {E} }}}+{\frac {1}{\|x-x_{-}\|_{\mathbb {E} }}}\right).

{\ displaystyle F_ {U} (x, y) = {\ frac {1} {2}} \ | y \ | _ {\ mathbb {E}} \ left ({\ frac {1} {\ | x- x _ {+} \ | _ {\ mathbb {E}}}} + {\ frac {1} {\ | x-x _ {-} \ | _ {\ mathbb {E}}}} \ right).}

This metric is also flat.

D. Hilbert introduced it in 1895 as a generalization of Lobachevsky's geometry. When the hypersurface $\partial U$ ${\ displaystyle \ partial U}$ is an ellipsoid, then we obtain the Lobachevsky geometry.

Funk metric

In 1930, Funk introduced an asymmetric metric. It is defined in the region bounded by a closed convex hypersurface and is also flat.

σ-metrics

Sufficient condition for flat metrics.

The first contribution to the solution of the IV problem of Hilbert was made by Hamel ^[3] . He proved the following statement.

Theorem . If the regular Finsler metric $F(x,y)=F(x_{1},....x_{n},y_{1},...,y_{n})$ ${\ displaystyle F (x, y) = F (x_ {1}, .... x_ {n}, y_ {1}, ..., y_ {n})}$ satisfies the condition

${\frac {\partial ^{2}F^{2}}{\partial x^{i}\partial y^{j}}}={\frac {\partial ^{2}F^{2}}{\partial x^{j}\partial y^{i}}},\,i,j=1..n,$ ${\ displaystyle {\ frac {\ partial ^ {2} F ^ {2}} {\ partial x ^ {i} \ partial y ^ {j}}} = {\ frac {\ partial ^ {2} F ^ { 2}} {\ partial x ^ {j} \ partial y ^ {i}}}, \, i, j = 1..n,}$

then it is flat.

Crofton Formula

Consider the set of oriented lines on the plane. The line is set by parameters $\rho ,\varphi ,$ ${\ displaystyle \ rho, \ varphi,}$ Where $\rho$ ${\ displaystyle \ rho}$ - the distance to the line from the origin, $\varphi$ ${\ displaystyle \ varphi}$ - the angle that forms a straight line with the axis Ox . Then the set of oriented lines is homeomorphic to a circular cylinder of unit radius, where the area element is $dS=d\rho d\varphi$ ${\ displaystyle dS = d \ rho d \ varphi}$ . Let be $\gamma$ ${\ displaystyle \ gamma}$ - rectifiable curve on the plane. Then its length

L={\frac {1}{4}}\iint \limits _{\Omega }n(\rho ,\varphi )dpd\varphi

{\ displaystyle L = {\ frac {1} {4}} \ iint \ limits _ {\ Omega} n (\ rho, \ varphi) dpd \ varphi}

,

Where $\Omega$ ${\ displaystyle \ Omega}$ - the set of lines that intersect this curve, $n(p,\varphi )$ ${\ displaystyle n (p, \ varphi)}$ - the number of intersections of a line with a curve. This was shown by M. Crofton in 1870.

A similar statement holds in the projective space ^[7] .

Measure Blaschke-Buseman

In 1966, G. Buseman, speaking at the International Mathematical Congress in Moscow, introduced a new class of flat metrics. G. Buseman introduced on the set of lines of the projective plane $RP^{2}$ ${\ displaystyle RP ^ {2}}$ completely additive non-negative measure $\sigma$ ${\ displaystyle \ sigma}$ which satisfies the following conditions:

$\sigma (\tau P)=0$ ${\ displaystyle \ sigma (\ tau P) = 0}$ where $\tau P$ ${\ displaystyle \ tau P}$ - the set of lines passing through the point P ;
$\sigma (\tau X)>0$ ${\ displaystyle \ sigma (\ tau X)> 0}$ where $\tau X$ ${\ displaystyle \ tau X}$ - the set of lines passing through some set X containing a line segment;
$\sigma (RP^{n})$ ${\ displaystyle \ sigma (RP ^ {n})}$ is finite.

If we consider $\sigma$ ${\ displaystyle \ sigma}$ -metric defined in an arbitrary convex region $\Omega$ ${\ displaystyle \ Omega}$ projective space $RP^{2}$ ${\ displaystyle RP ^ {2}}$ , then condition 3) is replaced by the requirement that for any set H , such that H is contained in $\Omega$ ${\ displaystyle \ Omega}$ , the closure of H does not intersect with the boundary $\Omega$ ${\ displaystyle \ Omega}$ , performed

\sigma (\pi H)<\infty

{\ displaystyle \ sigma (\ pi H) <\ infty}

^[8] .

Using such a measure is determined $\sigma$ ${\ displaystyle \ sigma}$ -metric in $RP^{2}$ ${\ displaystyle RP ^ {2}}$ :

|x,y|=\sigma \left(\tau [x,y]\right),

{\ displaystyle | x, y | = \ sigma \ left (\ tau [x, y] \ right),}

Where $\tau [x,y]$ ${\ displaystyle \ tau [x, y]}$ - a set of straight lines intersecting a segment $[x,y]$ ${\ displaystyle [x, y]}$ .

The triangle inequality for this metric follows from the Pasha theorem.

Theorem . $\sigma$ ${\ displaystyle \ sigma}$ -metric in $RP^{2}$ ${\ displaystyle RP ^ {2}}$ is a flat metric, that is, the geodesics in this metric are straight lines of the projective space.

But Buseman was far from thinking that $\sigma$ ${\ displaystyle \ sigma}$ -metrics exhaust all flat metrics. He wrote: “... Freedom in choosing metrics when specifying geodesics in the case of non-Riemannian metrics is so great that one can doubt whether there really is a convincing characteristic of all desarg spaces ...” ^[8] .

Two-dimensional case

Pogorelov's theorem

An amazing theorem proved in 1973 by A. V. Pogorelov ^[9] ^[10] proved to be surprising.

Theorem . Any two-dimensional continuous full flat metric is $\sigma$ ${\ displaystyle \ sigma}$ -metrics.

Thus, the IV Hilbert problem for the two-dimensional case is completely solved.

Other evidence

In 1976, R. B. Ambartsumian gave another proof of the IV problem of Hilbert ^[11] . His proof is connected with the fact that in the two-dimensional case the entire measure is restored by its values on the two-corners. And then it is defined on triangles in the same way as the area of a triangle on a sphere is specified. On non-degenerate triangles, it is positive since the triangle inequality holds, and then the measure is determined on all Borel sets. But this construction is not generalized by dimension. This is due to the third Hubert problem, which was solved by M. Den. In the two-dimensional case, equal polygons are equally spaced. In a larger dimension, as shown by M. Den, this is not true.

3D case

For the case n = 3, A. V. Pogorelov proved the following theorem

Theorem. Any three-dimensional regular continuous complete flat metric is $\sigma$ ${\ displaystyle \ sigma}$ -metrics.

However, in the three-dimensional case $\sigma$ ${\ displaystyle \ sigma}$ -measures can take both positive and negative values. The necessary and sufficient conditions for a regular metric given by a set function $\sigma$ ${\ displaystyle \ sigma}$ , was flat, are the following three conditions:

value $\sigma$ ${\ displaystyle \ sigma}$ on any plane equals zero;
value $\sigma$ ${\ displaystyle \ sigma}$ in any cone is non-negative;
value $\sigma$ ${\ displaystyle \ sigma}$ positive if the cone contains interior points.

In addition, AV Pogorelov showed that every complete continuous flat metric in the three-dimensional case is the limit of regular $\sigma$ ${\ displaystyle \ sigma}$ -metrics with uniform convergence in any compact subdomain of the domain of the given metric. Such metrics he called generic $\sigma$ ${\ displaystyle \ sigma}$ metrics

Thus, A. V. Pogorelov managed to prove that

Theorem. Every complete continuous flat metric in the three-dimensional case is $\sigma$ ${\ displaystyle \ sigma}$ -metrics in a generalized sense.

G. Buseman in a review of the translation of the book by A. V. Pogorelov `` The fourth problem of Hilbert wrote: "" In accordance with the spirit of the times, Hilbert limited himself to the dimensions n = 2, 3 . n = 2 and n> 2. The Pogorelov method works and for n> 3 only requires more technical details ^[12] . ”

Multidimensional case

The multidimensional case of the IV problem of Hilbert was studied by Z. I. Sabo. In 1986, he proved, as he himself writes, the generalized Pogorelov theorem: Theorem. Any n- dimensional desarg class space $C^{n+2},n>2$ ${\ displaystyle C ^ {n + 2}, n> 2}$ generated by the Blaschke-Busemann construction.

$\sigma$ ${\ displaystyle \ sigma}$ -measure that generates a flat measure, has the following properties:

$\sigma$ ${\ displaystyle \ sigma}$ The measure of the hyperplanes that pass through a fixed point is zero.
$\sigma$ ${\ displaystyle \ sigma}$ -measure of the set of hyperplanes that intersect two segments [x, y], [y, z] , where x, y, z are not collinear, is positive.

The same article provides an example of a flat metric that is not generated by the Blaschke-Busemann construction. ZI Sabo described all continuous flat metrics in the language of generalized functions ^[13] .

IV Hilbert's problem and bulging bodies

IV Hilbert's problem is also closely related to the properties of convex bodies. A convex polyhedron is called a zonotop if it is the sum (according to Minkowski) of segments. A convex body that is the limit of zonotopes in the Blaschke – Hausdorff metric is called a zonoid . For zonoids, the support function is represented as

h(x)=\int \limits _{S^{n-1}}|\langle x,u\rangle |\partial \sigma (u),

{\ displaystyle h (x) = \ int \ limits _ {S ^ {n-1}} | \ langle x, u \ rangle | \ partial \ sigma (u),}

Where $\sigma (u)$ ${\ displaystyle \ sigma (u)}$ Is an even positive Borel measure on a sphere $S^{n-1}$ ${\ displaystyle S ^ {n-1}}$ .

The Minkowski space is then and only then generated by the Blaschke-Busemann construction, when the supporting function of the indicatrix is given above, where $\sigma (u)$ ${\ displaystyle \ sigma (u)}$ Is an even not necessarily constant Borel measure ^[14] . Bodies bounded by such hypersurfaces are called generalized zonoids.

Octahedron $|x_{1}|+|x_{2}|+|x_{3}|\leqslant 1$ ${\ displaystyle | x_ {1} | + | x_ {2} | + | x_ {3} | \ leqslant 1}$ in Euclidean space $E^{3}$ ${\ displaystyle E ^ {3}}$ not a generalized zonoid. Then from the statement above it follows that the flat metric of Minkowski space with norm $\|x\|=\max\{|x_{1}|,|x_{2}|,|x_{3}|\}$ ${\ displaystyle \ | x \ | = \ max \ {| x_ {1} |, | x_ {2} |, | x_ {3} | \}}$ , not generated by the Blaschke-Busemann construction.

Generalizations IV of the Hilbert problem

The correspondence between flat n- dimensional Finsler metrics and special symplectic forms on a Grassmann manifold is found. $G(n+1,2)$ ${\ displaystyle G (n + 1,2)}$ at $E^{n+1}$ ${\ displaystyle E ^ {n + 1}}$ ^[15] .

Periodic solutions to the IV problem of Hilbert were considered:

Let (M, g) be a compact locally Euclidean Riemannian manifold. It is set $C^{2}$ ${\ displaystyle C ^ {2}}$ Finsler metric whose geodesics coincide with the metric g . Then the Finsler metric is the sum of the locally Minkow metric and the closed 1-form ^[16] .

Let (M, g) be a compact symmetric Riemannian space of rank greater than one. If F is symmetric $C^{2}$ ${\ displaystyle C ^ {2}}$ a Finsler metric whose geodesics coincide with the geodesic of the Riemannian metric g, then (M, F) is a symmetric Finsler space ^[16] .

Another presentation of the IV problem of Hubert is in the work of Payvi 2003 ^[17] .

Unresolved Tasks

Not solved IV problem of Hilbert for asymmetric distance.
An analogue of the last theorem is unknown for the case of rank-one symmetric spaces.
Describe metrics on $RP^{n}$ ${\ displaystyle RP ^ {n}}$ , for which k -planes minimize k- area (G. Buseman) ^[18] .

Literature

↑ D. Hilbert, Mathematische Probleme , Gottinger Nachrichten, 1900, 253–297
↑ G. Darboux, Lecons sur la theorie generale des surfaces , V.III, Paris, 1894.
↑ ¹ ² G. Hamel, Uber die Geometrien in denen die Geraden die Kurzesten sind , Math. Ann. 57 (1903), 221-264.
↑ E. Beltrami, Risoluzione del Problema: Riportare I punti di una superficie line line line line line ge ge ode geodetiche Ven Vengano rappresentate da linee rette , Annali di Matematica Pura ed Applicata, No. 7 (1865), 185—204
↑ H. Minkowski, Geometrie der Zahlen , Lpz.-B., 1953
↑ D. Hilbert, Uber die gerade Linie als kurzeste Verbindung zweier Punkte , Math. Ann., 46 (1895), 91-96
Sant LA Santalo, Integral geometry .- In: Studies in Global Geometry and Analysis (SS Chern, ed.), Washington, DC: Math. Asoc. Amer, 147–195
↑ ¹ ² G. Buseman, Geometry of Geodesics , Moscow, 1962.
↑ A.V. Pogorelov, Complete solution IV of the Hilbert problem , DAN USSR № 208, v.1 (1973), 46-49. AV Pogorelov, A complete solution of "Hilbert's fourth problem , Dokl. Acad. Nauk SSR, Vol. 208, No. 1 (1973), 48-52.
↑ A.V. Pogorelov, Hilbert's Fourth Problem . Ed. Science, 1974. English translation: AV Pogorelov, Hilbert's Fourth Problem , Scripta Series in Mathematics, Winston and Sons, 1979.
↑ RV Ambartzumian, A note on pseudo-metric on the plane , Z. Wahrscheinlichkeits theor. Verw. Geb 37 (1976), 145–155.
↑ H. Busemann, Review of: AV Pogorelov, Hilbert's fourth problem , Bull. Amer. Math Soc. (NS) Vol. 4, No. 1 (1981), 87-90.
↑ ZI Szabo, Hilbert's fourth problem I , Adv. Math 59 (1986), 185-301.
↑ R. Alexander, Zonoid theory and Hilbert fourth problem , Geom. Dedicata 28, No. 2 (1988), 199-211.
↑ JC Alvarez Paiva, Sympletic geometry and Hilbert fourth problem , J. Differ. Geom 69, No. 2 (2005), 353-378.
↑ ¹ ² JC Alvarez Pavia and J. Barbosa Gomes, Periodic Solutions of the Fourth Problem , 20 pp. arXiv: 1809.02783v1 [math.MG], 2018.
C JC Alvarez Paiva, Hilbert 4th MASS SELECTa: teaching and learning advanced undergraduate mathematics, ed. S. Katok et al., Providence, RI, AMS, (2003), 165-183.
↑ A. Papadopoulos, On Hilbert fourth problem , 1-43. Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, ed.), European Mathematical Society, IRMA Lectures in Mathematics and Theoretical Physics, No. 22 (2014), p. 460.

[1] D. Hilbert, Mathematische Probleme , Gottinger Nachrichten, 1900, 253–297

[2] G. Darboux, Lecons sur la theorie generale des surfaces , V.III, Paris, 1894.

[Hamel1903-3] ¹ ² G. Hamel, Uber die Geometrien in denen die Geraden die Kurzesten sind , Math. Ann. 57 (1903), 221-264.

[4] E. Beltrami, Risoluzione del Problema: Riportare I punti di una superficie line line line line line ge ge ode geodetiche Ven Vengano rappresentate da linee rette , Annali di Matematica Pura ed Applicata, No. 7 (1865), 185—204

[5] H. Minkowski, Geometrie der Zahlen , Lpz.-B., 1953

[6] D. Hilbert, Uber die gerade Linie als kurzeste Verbindung zweier Punkte , Math. Ann., 46 (1895), 91-96

[7] Sant LA Santalo, Integral geometry .- In: Studies in Global Geometry and Analysis (SS Chern, ed.), Washington, DC: Math. Asoc. Amer, 147–195

[Buseman1962-8] ¹ ² G. Buseman, Geometry of Geodesics , Moscow, 1962.

[9] A.V. Pogorelov, Complete solution IV of the Hilbert problem , DAN USSR № 208, v.1 (1973), 46-49. AV Pogorelov, A complete solution of "Hilbert's fourth problem , Dokl. Acad. Nauk SSR, Vol. 208, No. 1 (1973), 48-52.

[10] A.V. Pogorelov, Hilbert's Fourth Problem . Ed. Science, 1974. English translation: AV Pogorelov, Hilbert's Fourth Problem , Scripta Series in Mathematics, Winston and Sons, 1979.

[11] RV Ambartzumian, A note on pseudo-metric on the plane , Z. Wahrscheinlichkeits theor. Verw. Geb 37 (1976), 145–155.

[12] H. Busemann, Review of: AV Pogorelov, Hilbert's fourth problem , Bull. Amer. Math Soc. (NS) Vol. 4, No. 1 (1981), 87-90.

[13] ZI Szabo, Hilbert's fourth problem I , Adv. Math 59 (1986), 185-301.

[14] R. Alexander, Zonoid theory and Hilbert fourth problem , Geom. Dedicata 28, No. 2 (1988), 199-211.

[15] JC Alvarez Paiva, Sympletic geometry and Hilbert fourth problem , J. Differ. Geom 69, No. 2 (2005), 353-378.

[Paiva2018-16] ¹ ² JC Alvarez Pavia and J. Barbosa Gomes, Periodic Solutions of the Fourth Problem , 20 pp. arXiv: 1809.02783v1 [math.MG], 2018.

[17] C JC Alvarez Paiva, Hilbert 4th MASS SELECTa: teaching and learning advanced undergraduate mathematics, ed. S. Katok et al., Providence, RI, AMS, (2003), 165-183.

[Papadopoulos2004-18] A. Papadopoulos, On Hilbert fourth problem , 1-43. Handbook of Hilbert geometry (A. Papadopoulos and M. Troyanov, ed.), European Mathematical Society, IRMA Lectures in Mathematics and Theoretical Physics, No. 22 (2014), p. 460.