on hilbert ivth problemmath.sjtu.edu.cn/conference/bannai/2019/data/20190621a/slides.pdf · on...

On Hilbert IVth Problem

Marc Troyanov (EPFL)

SJTU, June 21, 2019

On Hilbert IVthProblem

Marc Troyanov(EPFL)

Introduction

Statement of the IVthproblem

Historical context

Early results

Busemann’sconstruction

The Finsler Viewpoint

Abstract

In 1900, Hilbert formulated his famous list of 23 problems that greatlyinfluenced mathematics throughout the 20th century. Ten among thoseproblems where presented at the second International Congress ofMathematicians held in Paris in August 1900.

Hilbert’s fourth problem can be formulated (in a somewhat modernlanguage) as the problem of describing all geometries in a domain of theEuclidean space or projective space for which the straight lines haveminimal length.

Various specific meaning to the problem have been proposed and severalpartial solution have been given, notably by H. Hamel, H. Busemann,Pogorelov and Ambartzumian.

In this talk, we will briefly present the historical context of Hilbert’s

fourth problem, describe its relation with Hilbert and Finsler geometries

and integral geometry. We will also describe some variant of the

problem and classes of solution.


Marc Troyanov(EPFL)

Introduction


Historical context

Early results



Plan

I Introduction: The Hilbert Problems.

I Historical context.

I Early Results

I The Busemann Construction

I The Finsler Viewpoint

I Generalization

I References


Marc Troyanov(EPFL)

Introduction


Historical context

Early results



Introduction: The Hilbert Problems.


Marc Troyanov(EPFL)

Introduction


Historical context

Early results



During the Second International Congress of Mathematicians heldin Paris in August 1900, David Hilbert delivered his famous lecturetitled The future problems of mathematics.

In this lecture he presented 10 open problems, together withcontext and mathematical comments. He then published anextended list of 23 problems first in the Swiss journalL’Enesignement Mathematiques then in the Bulletin of theAmerican Mathematical Society.


Marc Troyanov(EPFL)

Introduction


Historical context

Early results



The Hilbert problems present a wide variety of research area andthey greatly influenced mathematics during the 20th century. Hereare some of the problems:

Problem I. The continuum hypothesis (P. Cohen proved theCH to be independent of ZFC in 1963).

Problem II. Consistency of the axioms of arithmetic (K.Godelshowed the consistency cannot be proved in 1931)

Problem VIII. The Riemann hypothesis (yet unresolved).

Problem X. Algorithmic solvability of Diophantine equations(answered negatively by Matiyasevich in 1970)

Problem XIX. Are the solutions of regular problems in the calculusof variations always necessarily analytic? (solvedpositively by de Giorgi and Nash in 1957).


Marc Troyanov(EPFL)

Introduction


Historical context

Early results



By contrast to some of the other problems, Hilbert Problem IV iswritten as a long discussion:

Problem IV: Problem of the straight line as the shortestdistance between two points.

Another problem relating to the foundations of geometry is this: If fromamong the axioms necessary to establish ordinary Euclidean geometry,we exclude the axiom of parallels, or assume it as not satisfied, butretain all the other axioms, we obtain, as is well known, the geometry ofLobachevsky (hyperbolic geometry). We may therefore say that this is ageometry standing next to Euclidean geometry.

If we require further that that axiom be not satisfied whereby, of threepoints on a straight line, one and only one lies between the other two,we obtain Riemann’s (elliptic) geometry, so that this geometry appearsto be the next after Lobachevsky’s. [...]


Marc Troyanov(EPFL)

Introduction


Historical context

Early results



[...] The more general question now arises: Whether from othersuggestive standpoints geometries may not be devised which, with equalright, stand next to Euclidean geometry. Here I should like to directyour attention to a theorem which has, indeed, been employed by manyauthors as a definition of a straight line, viz., that the straight line isthe shortest distance between two points. The essential content of thisstatement reduces to the theorem of Euclid that in a triangle the sum oftwo sides is always greater than the third side – a theorem which, as iseasily seen, deals solely with elementary concepts, i. e., with such as arederived directly from the axioms, and is therefore more accessible tological investigation. Euclid proved this theorem, with the help of thetheorem of the exterior angle, on the basis of the congruence theorems.Now it is readily shown that this theorem of Euclid cannot be provedsolely on the basis of those congruence theorems which relate to theapplication of segments and angles, but that one of the theorems on thecongruence of triangles is necessary.

We are asking then, for a geometry in which all the axioms of ordinaryEuclidean geometry hold, and in particular all the congruence axiomsexcept the one of the congruence of triangles (or all except the theoremof the equality of the base angles in the isosceles triangle), and inwhich, besides, the proposition that in every triangle the sum of twosides is greater than the third is assumed as a particular axiom.


Marc Troyanov(EPFL)

Introduction


Historical context

Early results



One finds that such a geometry really exists and is no other than thatwhich Minkowski constructed in his book, Geometrie der Zahlen, andmade the basis of his arithmetical investigations. Minkowski’s istherefore also a geometry standing next to the ordinary Euclideangeometry; it is essentially characterized by the following stipulations:

1. The points which are at equal distances from a fixed point O lie on aconvex closed surface of the ordinary Euclidean space with O as acenter.2. Two segments are said to be equal when one can be carried to theother by a translation of the ordinary Euclidean space.

In Minkowski’s geometry the axiom of parallels also holds. By studyingthe theorem of the straight line as the shortest distance between twopoints, I arrived at a geometry in which the parallel axiom does nothold, while all other axioms of Minkowski’s geometry are satisfied. Thetheorem of the straight line as the shortest distance between two pointsand the essentially equivalent theorem of Euclid about the sides of atriangle, play an important part not only in number theory but also inthe theory of surfaces and in the calculus of variations.


Marc Troyanov(EPFL)

Introduction


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For this reason, and because I believe that the thorough investigation ofthe conditions for the validity of this theorem will throw a new lightupon the idea of distance, as well as upon other elementary ideas, e.g.,upon the idea of the plane, and the possibility of its definition by meansof the idea of the straight line, the construction and systematictreatment of the geometries here possible seems to me desirable.


Marc Troyanov(EPFL)

Introduction


Historical context

Early results



You may rightfully be confused by this text: What exactly isHilbert talking about? What is the precise statement of the fourthproblem? and why is it stated in such a complicate language?

To give a short answer, we can refer to Wikipedia, where theproblem is stated as

“Construct all metrics where lines are geodesics”.

Wikipedia also adds the following comment:

“Too vague to be stated resolved or not” [sic.]

On these points I dare say that Wikipedia is a bit too vague.


Marc Troyanov(EPFL)

Introduction


Historical context

Early results



A brief historical context


Marc Troyanov(EPFL)

Introduction


Historical context

Early results



To better grasp the problem it is useful to have in mind some ofthe key aspects of the history of Geometry in the XIXth century.

In the 1820’s three major developments arised that greatlyinfluenced the development of Geometry. These are

I The revival of Projective Geometry by Jean-Victor Ponceletand Charles Julien Brianchon (and later Karl von Staudt).

I The discovery of Non Euclidean Geometry by NikolaiIvanovich Lobachevsky and Janos Bolyai.

I The work on Carl Friedriech Gauss on the differentialgeometry of surfaces (in particular the notion of intrinsiccurvature).

It will take several decades until the deep relations between thesethree sides of geometry will be clarified.


Marc Troyanov(EPFL)

Introduction


Historical context

Early results



In Arthur Cayley’s A sixth Memoir upon Quantics (1859), theauthor associates to conic curve (which he calls the absolute) adistance function in the projective plane such that for any threepoints P1,P2,P3 in that order on a projective line, we have

d(P1,P2) + d(P2,P3) = d(P1,P3)

The construction is rather obscure, but a few years later FelixKlein proved that the Cayley metric can be expressed in terms ofcross ratios, a quantity that was well known in projectivegeometry. He also observed that when the conic is an ellipse theobtained geometry is Lobatchevsky’s non Euclidean geometry.

Eugenio Beltrami independently reached the same conclusion andthe geometry thus constructed is called the Cayley-Beltrami-Kleinmodel of hyperbolic geometry (or simply the Klein Model to makeit short).


Marc Troyanov(EPFL)

Introduction


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Early results



P′

Q′

P

Q

d(P,Q) =1

2log

(|P ′ − Q||P ′ − P|

· |Q′ − P|

|Q ′ − Q|

)


Marc Troyanov(EPFL)

Introduction


Historical context

Early results



Let us observe the following properties of this formula:

I The metric is only defined for points inside the ellipse.

I The metric is complete (if a sequence of points Qjconverges to the boundary of the elliptic domain, thend(P,Qj)→∞)

I The metric is projective, that is if P2 belongs to the segment[P,P3] then d(P1,P2) + d(P2,P3) = d(P1,P3).


Marc Troyanov(EPFL)

Introduction


Historical context

Early results



In 1865, Beltrami approached the problem from the point of viewof differential geometry. Namely he asked: Which surfaces can berepresented on the plane in such a way that the geodesics of thesurface are mapped on straight lines?

An example was well known to cartographers since the Antiquity:the Gnomonic projection is the map that projects a sphere from itscenter to one of its tangent plane. It clearly maps the geodesics ofthe sphere (which are the great circles) to straight lines.


Marc Troyanov(EPFL)

Introduction


Historical context

Early results




Marc Troyanov(EPFL)

Introduction


Historical context

Early results



Beltrami proved the following

Theorem A surface can locally be mapped geodesically on theEuclidean plane if and only if the surface has constant Gaussiancurvature.

He also observed that the planeof hyperbolic geometry is lo-cally isometric to a surface ofconstant Gaussian CurvatureK = −1.That surface is known as Bel-trami’s pseudo-sphere and isobtained as the surface of rev-olution of the tractrix curve.


Marc Troyanov(EPFL)

Introduction


Historical context

Early results



Beltrami’s Theorem has later been generalized to Riemannianmanifolds of all dimension. A modern proof can be found in thepaper by Vladimir Matveev, Geometric explanation of the BeltramiTheorem. Int. J. Geom. Methods Mod. Phys. 3 (2006), no. 3,623–629.


Marc Troyanov(EPFL)

Introduction


Historical context

Early results



In 1895, D. Hilbert sent a letter to Klein which contained twoparts. We first describe the second part. In that part Hilbertobserved that for an arbitrary bounded convex domain in Rn, theformula

d(P,Q) =1

2log

(|P ′ − Q||P ′ − P|

· |Q′ − P|

|Q ′ − Q|

),

where P ′ and Q ′ are the intersection of the line through P and Qwith the boundary of the domain determine a metric on thatdomain.

P′

Q′

P

Q

Furthermore the metric is complete and projective:

P2 ∈ [P1,P3]⇒ d(P1,P2) + d(P2,P3) = d(P1,P3).


Marc Troyanov(EPFL)

Introduction


Historical context

Early results



The proof of the triangle inequality is rather subtle though it usesonly classical Euclidean geometry. Hilbert also gave necessary andsufficient condition for the metric to be strictly projective, that is

P2 ∈ [P1,P3] ⇔ d(P1,P2) + d(P2,P3) = d(P1,P3).

A sufficient (but not necessary) condition is that the domain bestrictly convex.

As a point of terminology let us mention that we nowadays call aHilbert Geometry a pair (Ω, d), where Ω ⊂ Rn and d is theHilbert metric in that domain.


Marc Troyanov(EPFL)

Introduction


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Early results



Minkowski GeometryIn 1896, Hermann Minkowski published his famous book“Geometrie der Zahlen ” (Geometry of Numbers). In that bookMinkowski observed that to any bounded open symmetric domainΩ in Rn one can associate a distance ρ on the whole of Rn whichis invariant by translation and for which every sphere of radius 1 isa translate of Ω.

The distance is (clearly) given by

ρ(P,Q) = ‖Q − P‖

where ‖ ‖ is the norm whose unit sphere is given by Ω (assumingits center is the origin of Rn). Observe however that this simpleidea was completely new to the mathematical community.

Observe also that the metric ρ is projective and complete. Themetric is strictly projective if and only if Ω is strictly convex.


Marc Troyanov(EPFL)

Introduction


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Early results



We nowadays call a Minkowsi Geometry the pair (E , ‖ ‖) where Eis a finite dimensional Banach space and ‖ ‖ is a norm on E .

Thus Minkowski geometries are generalizations of Euclideangeometry (the parallel axiom holds) while Hilbert geometry is ageneralization of Hyperbolic geometry (Lobatchevsky Geometry).


Marc Troyanov(EPFL)

Introduction


Historical context

Early results



Calculus of Variation In 1894, Gaston Darboux considered theproblem of determining for which Lagrangians F = F (x , v) on Rn

the straight lines are the extremals of the action integral

S(γ) =

∫γ

F (γ(t), γ(t))dt,

and he obtained some partial result (later generalized in particularby Hammel).Darboux saw this question as an extension of Beltrami’s Theorem.Nowadays these kind of questions are refered to as Inverseproblems in the calculus of variation.


Marc Troyanov(EPFL)

Introduction


Historical context

Early results



The Foundation of GeometryThe last two decades of the XIXth century saw the development offormal logic and set theory and was a period of intensequestioning on the general foundation of mathematics.

In particular the axioms of Euclidean geometry as written down inEuclid’s Elements where clearly perceived to be insufficient for arigorous foundation of the subject.A new approach has been proposed by Peano and Moritz Pashbased on the notion of betweeness. Namely these authors assumeas primitive notions the points (denoted by A,B,C ...) and theternary relation of betweeness A ∗ B ∗ C which can be read as ”Blies between A and C”, or ”B belongs to the segment [A,B]”

If A and B are distinct points, then the line through A and B isthe set of points P such that A ∗B ∗P, P ∗A ∗B ∗C or A ∗P ∗B.


Marc Troyanov(EPFL)

Introduction


Historical context

Early results



A list of axioms regulates the basic properties of lines (these arethe incidence axioms) and the properties of the betweenessrelation (these are the order axioms). Let us just quote the famousPasch Axiom:

Axiom of Pasch. If ABC is a triangle and P, Q are points suchthat P ∗ C ∗ B and A ∗ Q ∗ C then there exists a point R on theline PQ for which A ∗ R ∗ B.

A

B

C

P

R

Q

Intuitively, Pasch’s axiom say that if a line meets the interior ofone side of the triangle then it meets another side.

Note: The formulation in Pasch’s work was slightly different. In this

form the axiom is due to Veblen and Young.


Marc Troyanov(EPFL)

Introduction


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Early results



Hilbert’s axiomsIn 1899, Hilbert published his famous book Grundlagen derGeometrie (The Foundation of Geometry) where he lays down asolid axiomatic foundation of (three dimensional) Euclideangeometry.His foundation is based on 3 primitive notions (the points, the linesand the plane) and comprises 20 axioms organized in 5 groups:

1. Axioms of incidence.

2. Axioms of order.

3. Axioms of congruence.

4. The parallel postulate.

5. Axiom of continuity.

The first two groups essentially reproduce the Peano-Pasch ordergeometry, including the Pasch Axiom.

The third group contains 3 axioms on congruence of segments and3 axioms on congruence of angles.


Marc Troyanov(EPFL)

Introduction


Historical context

Early results



Based on these axioms, Hilbert successfully reproduce all results ofEuclidean geometry. Including the following Theorem:

Euclid’s Elements, Book 1,Proposition 20 In any triangle, the sum of two sides exceeds thethird side.

This is the triangle inequality. Note however that the notion oflength has not appeared yet in the construction. The abovetheorem is to be interpreted in terms of congruence and order (i.e.we are comparing segments, not real numbers).

The fact that Euclidean space can be seen as a metric space(isometric to R3) is a Theorem appearing rather late in Hilbert’sconstruction.


Marc Troyanov(EPFL)

Introduction


Historical context

Early results



Returning to Hilbert Problem IV, let us recall that Hilbert suggestto consider a geometry in which the Parallel Postulate as well asthe congruence axioms for angles are dropped, and the triangleinequality is now assumed as an axiom.

And he writes: the construction and systematic treatment of thegeometries here possible seems to me desirable.


Marc Troyanov(EPFL)

Introduction


Historical context

Early results



Hilbert seems to have been aware of the following fact:

Fundamental Theorem of Convex Geometry An orderGeometry satisfying the congruence axioms for the segments andthe continuity axiom is isomorphic to the geometry of a convexdomain in Rn or RPn.

Note that such a geometry is isomorphic to Rn if and only if theparallel axiom holds.


Marc Troyanov(EPFL)

Introduction


Historical context

Early results



Early Results


Marc Troyanov(EPFL)

Introduction


Historical context

Early results



Metric SpacesThe modern notion of abstract metric spaces has been defined byM. Frechet in 1906. Taking into account the previous Theorem itis then natural to reformulate Hilbert Problem IV as follows:

Problem. Given a convex domain Ω in Rn or RPn describe all themetrics d : Ω× Ω→ R that are complete and projective:

P2 ∈ [P1,P3]⇒ d(P1,P2) + d(P2,P3) = d(P1,P3).

or strictly projective:

P2 ∈ [P1,P3] ⇔ d(P1,P2) + d(P2,P3) = d(P1,P3).


Marc Troyanov(EPFL)

Introduction


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Early results



The first major result on Hilbert Problem IV is due to GeorgHamel, who was a student of Hilbert. He proved in his thesis in1903 the following remarkable result:

TheoremLet Ω ⊂ RPn be a convex subset of the projective space and d astrictly projective metric on Ω. Assume that closed d-balls arecompact. Then one of the following cases hold:

1. Ω = RPn and and all great circles are geodesics of equallengths;

2. Ω is projectively equivalent to a convex domain in Rn and theintersection of Ω with affine lines are maximal geodesics ofinfinite length.


Marc Troyanov(EPFL)

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H. Busemann’s viewpoint and result


Marc Troyanov(EPFL)

Introduction


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Early results



Note that Hamel’s Theorem describes a part of the structure ofthe solutions to Hilbert Problem IV, but it does not address theexistence.Busemann observed the following

Proposition. Every convex domain Ω ⊂ RPn admits a metric dthat is complete, strictly projective and for which the closed ballsare compact.

Proof. If Ω = RPn or Ω ' Rn then the standard spherical or Euclideanmetric is a solution.

If Ω ⊂ Rn is a strictly convex domain, then the Hilbert metric is asolution.If Ω ⊂ Rn is a convex domain but not strictly convex, then a solution isgiven by

d(x , y) = dH(x , y) + |y − x |,

where dH is the Hilbert metric in Ω and |y − x | is the Euclidean metric.


Marc Troyanov(EPFL)

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Note that one can replace the Euclidean norm by any strictlyconvex convex norm in the previous argument.

More generally, suppose that d1 and d2 are two projective metricson Ω, then d = α1d1 + α2d2 is again a projective metric for anyα1, α2 > 0. Furthermore:

i.) If d1 or d2 is strictly projective, then d is strictly projective.

ii.) If d1 or d2 is complete, then d is complete.

It thus seems hopeless to classify all projective metrics on a convexdomain...

Note however that the above remark means that the projectivemetrics form a convex cone in the space of all functionsΩ× Ω→ R.


Marc Troyanov(EPFL)

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It is often useful in studying metric spaces to also consider variousgeneralization of the notion of distance function. The weakestnotion is the following

Definition. A weak metric on a set X is a functionδ : X × X → R such that for all x ,y and z in X we have

(a) δ(x , y) ≥ 0 and δ(x , x) = 0.

(b) δ(x .z) ≤ δ(x , y) + δ(y , z).


Marc Troyanov(EPFL)

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Here is a simple example: Let Ω be a convex subset of Rn andH ⊂ Rn be a hyperplane. The wall metric associated to H in Ω isthe weak metric wH defined by

wH(x , y) =

1 if H separates x from y ,

0 otherwise.

This is clearly a projective weak metric:

y ∈ [x , z ] ⇒ wH(x , z) = wH(x , y) + wH(y , z).

Remark. For three points not on a line the triangle inequality forthe wall metric is equivalent to Pasch’s axiom applied to the planecontaining those three points.


Marc Troyanov(EPFL)

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The wall metric of a hyperplane is highly degenerate, but if weconsider a large (finite) family of hyperplanesH1,H2, . . . ,Hm ⊂ Rn, then these hyperplanes separate Ω inmultiple chambers and the weak distance

δ(x , y) =m∑i=1

wHi (x , y)

counts the number of walls the segment [x , y ] is crossing.

y

x


Marc Troyanov(EPFL)

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Note that the distance between two points in the same chambersis zero and the distance between two arbitrary points depends onlyon the chamber to which they belong.

One may include a cost µi ≥ 0 to traversing the wall Hi .In that case the associated weak distance to the system(H1, µi ), . . . (Hm, µm) is given by

δ(x , y) =m∑i=1

µiwHi (x , y (1)

).This is again a projective weak metric with the same properties,we may call the metric (1) a multiwall metric.


Marc Troyanov(EPFL)

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To obtain a non degenerate metric from this procedure, one needsto replace the previous finite sum with an integral. We thusconsider metrics of the type

d(x , y) =

∫HwH(x , y)dµ(H) (2)

Where H is the manifold of all hyperplanes in Rn and µ is a Borelmeasure on H.

Note that formula (2) can simply be rewritten as

d(x , y) = µ(H ∈ H | H separates x from y)


Marc Troyanov(EPFL)

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Theorem [H. Busemann]Let Ω be a convex domain in R2 and µ be a (non negative) Radonmeasure on the manifold HΩ of hyperplanes meeting Ω.Assume the following properties

(i.) µ(H ∈ HΩ | x ∈ H) = 0 for any point x ∈ Ω.

(ii.) µ(H ∈ HΩ | H ∩ [x , y ] 6= ∅) > 0 for any non degeneratesegment [x , y ] ⊂ Ω.

Then the formula

d(x , y) = µ(H ∈ HΩ | H ∩ [x , y ] 6= ∅) (3)

defines a non degenerate projective metric that is continuous withrespect to the topology of R2.


Marc Troyanov(EPFL)

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Furtermore, we have the following

Proposition 1. [Crofton Formula.]The length of any rectifiable curve C in Ω for the metric d is givenby

`(C ) = µ(H ∈ HΩ | H ∩ C 6= ∅)

Proposition 2. If Ω = Rn and the measure µ is invariant underthe action on H of the group of isometries of Rn, then the metricd is a multiple of the Euclidean metric.

Likewise if Ω = Bn is the unit ball and the measure µ is invariantunder the action on HBn of the group of projective tranformationsof Bn, then the metric d is a multiple of the Hyperbolic metric.


Marc Troyanov(EPFL)

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Let us examine under which condition on the measure µ themetric is strictly projective.Let us denote by H[x,y ] the set of hyperplanes metting thesegment [x , y ]. Then we have from Pasch’s axiom:

H[x,z] ∪H[z,y ] = (H[x,z] ∩H[z,y ]) ∪H[x,y ]

the latter being a disjoint union.Therefore

µ(H[x,z] ∪H[z,y ]) = µ(H[x,y ]) + µ(H[x,z] ∩H[z,y ])


Marc Troyanov(EPFL)

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The last equation can be written as

d(x , z) + d(z , y) = d(x , y) + µ(H[x,z] ∩H[z,y ])

Therefore we have

Proposition The metric associated to the measure µ on HΩ isstrictly projective if and only if for any triple x , y , z of non alignedpoints, the measure of the set of hyperplanes crossing both [x , z ]and [z , y ] is positive:

µ(H[x,z] ∩H[z,y ]) > 0.


Marc Troyanov(EPFL)

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In the converse direction we have the following result:

TheoremLet d be a continuous projective metric in a convex domain Ω ofthe plane R2.Then there exists a unique Radon measure on HΩ that satisfies (i)and (ii) and such that

d(x , y) = µ(H ∈ HΩ | H ∩ [x , y ] 6= ∅)


Marc Troyanov(EPFL)

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This result has been obtained in the 1970’s independently by A. V.Pogorelov, R. V. Ambartzumian and R. Alexander.

The idea of the construction of Ambartzumian is to consider forall non aligned triples x , y , z in Ω the set

Ux,y ,z = H[x,z] ∩H[z,y ]

of lines crossing both [x , z ] and [z , y ]. For those sets he thendefines µ to be

µ(Ux,y ,z) = d(x , z) + d(z , y)− d(x , y)

From the fact that the metric is projective, one sees that this setfunction is additive. We then extends this set function to anypolygonal region and then, by standard measure theoreticalarguments µ defines a Borel regular measure on Ω.


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The theorem still holds in higher dimension, but the constructionproduces a signed measure µ (that is it can take negative values).However we still have

µ(H[x,y ]) > 0 and µ(Ux,y ,z) > 0

for any non degenerate triples.


Marc Troyanov(EPFL)

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A quick view on the Finslerian viewpoint


Marc Troyanov(EPFL)

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If the measure µ is sufficiently regular, then the metric d isFinslerian. Let us quote two results in this direction.

Theorem (S. Ivanov 2009) If an arbitrary metric d is the planeis Lipschitz compared to the Euclidean metric, then it is a weakFinslerian metric: There exists an upper semicontinuous functionF : Ω×Rn → R such that F (x , ·) is a norm for any x ∈ Ω and thedistance d(x , y) is the infimum of the length of all curves γ joiningthem, where the length is defined as

`(γ) =

∫ t1

t0

F (γ(t), γ(t)dt


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In dimension 2 we have

Theorem (Pogorelov) Let d be a metric in a convex domainΩ ⊂ R2 be obtained from Busemann’s construction from a smoothmeasure dµ = fdA, then is Finslerian and the Finsler function Fcan be explicitely constructed from the function f .

Some 10 years ago, Juan Carlos Alvarez Paiva reinterpretedPogorelov’s result in the context of symplectic geometry andapplied the method to the analysis of Finsler metrics withprescribed geodesics.


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We conclude with what is perhaps the main result on theFinslerian theory of projectively flat metrics:

Theorem (Funk-Berwald, 1930’s) Let Ω be a bounded smoothand strongly convex domain. Then the Hilbert metric is theunique complete projective metric of constant flag curvature −1.


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J. C. Alvarez Paiva, Symplectic geometry and Hilbert’s fourthproblem. J. Differential Geom. 69 (2005).

R. V. Ambartzumian, A note on pseudo-metrics on the plane.Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 1976.

R. Alexander, Planes for which the lines are the shortest pathsbetween points. Illinois J. Math. 22 (1978), no. 2, 177–190.

Handbook of Hilbert Geometry Papadopoulos, Troyanov (Eds)

A. V. Pogorelov, Hilbert’s fourth problem. Wiley, 1979European Math. Soc Pblishing House, 2014.

Zhongmin Shen, Projectively flat Finsler metrics of constantflag curvature. Trans. Amer. Math. (2003).


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Xie Xie

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