On Hilbert IVth Problem Marc Troyanov (EPFL) SJTU, June 21, 2019 - - PowerPoint PPT Presentation

On Hilbert IVth Problem

Marc Troyanov (EPFL) SJTU, June 21, 2019

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

Abstract

In 1900, Hilbert formulated his famous list of 23 problems that greatly influenced mathematics throughout the 20th century. Ten among those problems where presented at the second International Congress of Mathematicians held in Paris in August 1900. Hilbert’s fourth problem can be formulated (in a somewhat modern language) as the problem of describing all geometries in a domain of the Euclidean space or projective space for which the straight lines have minimal length. Various specific meaning to the problem have been proposed and several partial 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.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

Plan ◮ Introduction: The Hilbert Problems. ◮ Historical context. ◮ Early Results ◮ The Busemann Construction ◮ The Finsler Viewpoint ◮ Generalization ◮ References

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

Introduction: The Hilbert Problems.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

During the Second International Congress of Mathematicians held in Paris in August 1900, David Hilbert delivered his famous lecture titled The future problems of mathematics. In this lecture he presented 10 open problems, together with context and mathematical comments. He then published an extended list of 23 problems first in the Swiss journal L’Enesignement Math´ ematiques then in the Bulletin of the American Mathematical Society.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

The Hilbert problems present a wide variety of research area and they greatly influenced mathematics during the 20th century. Here are some of the problems: Problem I. The continuum hypothesis (P. Cohen proved the CH to be independent of ZFC in 1963). Problem II. Consistency of the axioms of arithmetic (K.G¨

• del

showed 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 calculus

• f variations always necessarily analytic? (solved

positively by de Giorgi and Nash in 1957).

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

By contrast to some of the other problems, Hilbert Problem IV is written as a long discussion: Problem IV: Problem of the straight line as the shortest distance between two points.

Another problem relating to the foundations of geometry is this: If from among the axioms necessary to establish ordinary Euclidean geometry, we exclude the axiom of parallels, or assume it as not satisfied, but retain all the other axioms, we obtain, as is well known, the geometry of Lobachevsky (hyperbolic geometry). We may therefore say that this is a geometry standing next to Euclidean geometry. If we require further that that axiom be not satisfied whereby, of three points on a straight line, one and only one lies between the other two, we obtain Riemann’s (elliptic) geometry, so that this geometry appears to be the next after Lobachevsky’s. [...]

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

[...] The more general question now arises: Whether from other suggestive standpoints geometries may not be devised which, with equal right, stand next to Euclidean geometry. Here I should like to direct your attention to a theorem which has, indeed, been employed by many authors as a definition of a straight line, viz., that the straight line is the shortest distance between two points. The essential content of this statement reduces to the theorem of Euclid that in a triangle the sum of two sides is always greater than the third side – a theorem which, as is easily seen, deals solely with elementary concepts, i. e., with such as are derived directly from the axioms, and is therefore more accessible to logical investigation. Euclid proved this theorem, with the help of the theorem of the exterior angle, on the basis of the congruence theorems. Now it is readily shown that this theorem of Euclid cannot be proved solely on the basis of those congruence theorems which relate to the application of segments and angles, but that one of the theorems on the congruence of triangles is necessary. We are asking then, for a geometry in which all the axioms of ordinary Euclidean geometry hold, and in particular all the congruence axioms except the one of the congruence of triangles (or all except the theorem

• f the equality of the base angles in the isosceles triangle), and in

which, besides, the proposition that in every triangle the sum of two sides is greater than the third is assumed as a particular axiom.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

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

• 1. The points which are at equal distances from a fixed point O lie on a

convex closed surface of the ordinary Euclidean space with O as a center.

• 2. Two segments are said to be equal when one can be carried to the
• ther by a translation of the ordinary Euclidean space.

In Minkowski’s geometry the axiom of parallels also holds. By studying the theorem of the straight line as the shortest distance between two points, I arrived at a geometry in which the parallel axiom does not hold, while all other axioms of Minkowski’s geometry are satisfied. 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.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

For this reason, and because I believe that the thorough investigation of the conditions for the validity of this theorem will throw a new light upon 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 means

• f the idea of the straight line, the construction and systematic

treatment of the geometries here possible seems to me desirable.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

You may rightfully be confused by this text: What exactly is Hilbert talking about? What is the precise statement of the fourth problem? and why is it stated in such a complicate language? To give a short answer, we can refer to Wikipedia, where the problem 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.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

A brief historical context

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

To better grasp the problem it is useful to have in mind some of the key aspects of the history of Geometry in the XIXth century. In the 1820’s three major developments arised that greatly influenced the development of Geometry. These are ◮ The revival of Projective Geometry by Jean-Victor Poncelet and Charles Julien Brianchon (and later Karl von Staudt). ◮ The discovery of Non Euclidean Geometry by Nikolai Ivanovich Lobachevsky and J´ anos Bolyai. ◮ The work on Carl Friedriech Gauss on the differential geometry of surfaces (in particular the notion of intrinsic curvature). It will take several decades until the deep relations between these three sides of geometry will be clarified.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

In Arthur Cayley’s A sixth Memoir upon Quantics (1859), the author associates to conic curve (which he calls the absolute) a distance function in the projective plane such that for any three points 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 Felix Klein proved that the Cayley metric can be expressed in terms of cross ratios, a quantity that was well known in projective

• geometry. He also observed that when the conic is an ellipse the
• btained geometry is Lobatchevsky’s non Euclidean geometry.

Eugenio Beltrami independently reached the same conclusion and the geometry thus constructed is called the Cayley-Beltrami-Klein model of hyperbolic geometry (or simply the Klein Model to make it short).

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

P′ Q′ P Q

d(P, Q) = 1 2 log |P′ − Q| |P′ − P| · |Q′ − P| |Q′ − Q|

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

Let us observe the following properties of this formula: ◮ The metric is only defined for points inside the ellipse. ◮ The metric is complete (if a sequence of points {Qj} converges to the boundary of the elliptic domain, then d(P, Qj) → ∞) ◮ The metric is projective, that is if P2 belongs to the segment [P,P3] then d(P1, P2) + d(P2, P3) = d(P1, P3).

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

In 1865, Beltrami approached the problem from the point of view

• f differential geometry. Namely he asked: Which surfaces can be

represented on the plane in such a way that the geodesics of the surface 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 its center to one of its tangent plane. It clearly maps the geodesics of the sphere (which are the great circles) to straight lines.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

Beltrami proved the following Theorem A surface can locally be mapped geodesically on the Euclidean plane if and only if the surface has constant Gaussian curvature. He also observed that the plane

• f hyperbolic geometry is lo-

cally isometric to a surface of constant Gaussian Curvature K = −1. That surface is known as Bel- trami’s pseudo-sphere and is

• btained as the surface of rev-
• lution of the tractrix curve.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

Beltrami’s Theorem has later been generalized to Riemannian manifolds of all dimension. A modern proof can be found in the paper by Vladimir Matveev, Geometric explanation of the Beltrami

• Theorem. Int. J. Geom. Methods Mod. Phys. 3 (2006), no. 3,

623–629.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

In 1895, D. Hilbert sent a letter to Klein which contained two

• parts. We first describe the second part.

In that part Hilbert

• bserved that for an arbitrary bounded convex domain in Rn, the

formula d(P, Q) = 1 2 log |P′ − Q| |P′ − P| · |Q′ − P| |Q′ − Q|

• ,

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

P′ Q′ P Q

Furthermore the metric is complete and projective: P2 ∈ [P1, P3] ⇒ d(P1, P2) + d(P2, P3) = d(P1, P3).

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

The proof of the triangle inequality is rather subtle though it uses

• nly classical Euclidean geometry. Hilbert also gave necessary and

sufficient 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 be strictly convex. As a point of terminology let us mention that we nowadays call a Hilbert Geometry a pair (Ω, d), where Ω ⊂ Rn and d is the Hilbert metric in that domain.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

Minkowski Geometry In 1896, Hermann Minkowski published his famous book “Geometrie der Zahlen ” (Geometry of Numbers). In that book Minkowski observed that to any bounded open symmetric domain Ω in Rn one can associate a distance ρ on the whole of Rn which is invariant by translation and for which every sphere of radius 1 is a 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 (assuming its center is the origin of Rn). Observe however that this simple idea was completely new to the mathematical community. Observe also that the metric ρ is projective and complete. The metric is strictly projective if and only if Ω is strictly convex.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

We nowadays call a Minkowsi Geometry the pair (E, ) where E is a finite dimensional Banach space and is a norm on E. Thus Minkowski geometries are generalizations of Euclidean geometry (the parallel axiom holds) while Hilbert geometry is a generalization of Hyperbolic geometry (Lobatchevsky Geometry).

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

Calculus of Variation In 1894, Gaston Darboux considered the problem 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 particular by Hammel). Darboux saw this question as an extension of Beltrami’s Theorem. Nowadays these kind of questions are refered to as Inverse problems in the calculus of variation.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

The Foundation of Geometry The last two decades of the XIXth century saw the development of formal logic and set theory and was a period of intense questioning on the general foundation of mathematics. In particular the axioms of Euclidean geometry as written down in Euclid’s Elements where clearly perceived to be insufficient for a rigorous foundation of the subject. A new approach has been proposed by Peano and Moritz Pash based on the notion of betweeness. Namely these authors assume as primitive notions the points (denoted by A, B, C...) and the ternary relation of betweeness A ∗ B ∗ C which can be read as ”B lies 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 is the set of points P such that A ∗ B ∗ P, P ∗ A ∗ B ∗ C or A ∗ P ∗ B.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

A list of axioms regulates the basic properties of lines (these are the incidence axioms) and the properties of the betweeness relation (these are the order axioms). Let us just quote the famous Pasch Axiom: Axiom of Pasch. If ABC is a triangle and P, Q are points such that P ∗ C ∗ B and A ∗ Q ∗ C then there exists a point R on the line PQ for which A ∗ R ∗ B.

A B C P R Q

Intuitively, Pasch’s axiom say that if a line meets the interior of

• ne 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.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

Hilbert’s axioms In 1899, Hilbert published his famous book Grundlagen der Geometrie (The Foundation of Geometry) where he lays down a solid axiomatic foundation of (three dimensional) Euclidean geometry. His foundation is based on 3 primitive notions (the points, the lines and 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 order geometry, including the Pasch Axiom. The third group contains 3 axioms on congruence of segments and 3 axioms on congruence of angles.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

Based on these axioms, Hilbert successfully reproduce all results of Euclidean geometry. Including the following Theorem: Euclid’s Elements, Book 1, Proposition 20 In any triangle, the sum of two sides exceeds the third side. This is the triangle inequality. Note however that the notion of length has not appeared yet in the construction. The above theorem 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’s construction.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

Returning to Hilbert Problem IV, let us recall that Hilbert suggest to consider a geometry in which the Parallel Postulate as well as the congruence axioms for angles are dropped, and the triangle inequality is now assumed as an axiom. And he writes: the construction and systematic treatment of the geometries here possible seems to me desirable.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

Hilbert seems to have been aware of the following fact: Fundamental Theorem of Convex Geometry An order Geometry satisfying the congruence axioms for the segments and the continuity axiom is isomorphic to the geometry of a convex domain in Rn or RPn. Note that such a geometry is isomorphic to Rn if and only if the parallel axiom holds.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

Early Results

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

Metric Spaces The modern notion of abstract metric spaces has been defined by

• M. Frechet in 1906. Taking into account the previous Theorem it

is then natural to reformulate Hilbert Problem IV as follows:

• Problem. Given a convex domain Ω in Rn or RPn describe all the

metrics d : Ω × Ω → R that are complete and projective: P2 ∈ [P1, P3] ⇒ d(P1, P2) + d(P2, P3) = d(P1, P3).

• r strictly projective:

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

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

The first major result on Hilbert Problem IV is due to Georg Hamel, who was a student of Hilbert. He proved in his thesis in 1903 the following remarkable result:

Theorem

Let Ω ⊂ RPn be a convex subset of the projective space and d a strictly projective metric on Ω. Assume that closed d-balls are

• compact. Then one of the following cases hold:
• 1. Ω = RPn and and all great circles are geodesics of equal

lengths;

• 2. Ω is projectively equivalent to a convex domain in Rn and the

intersection of Ω with affine lines are maximal geodesics of infinite length.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

• H. Busemann’s viewpoint and result

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

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

• Proposition. Every convex domain Ω ⊂ RPn admits a metric d

that is complete, strictly projective and for which the closed balls are compact.

• Proof. If Ω = RPn or Ω ≃ Rn then the standard spherical or Euclidean

metric is a solution. If Ω ⊂ Rn is a strictly convex domain, then the Hilbert metric is a solution. If Ω ⊂ Rn is a convex domain but not strictly convex, then a solution is given by d(x, y) = dH(x, y) + |y − x|, where dH is the Hilbert metric in Ω and |y − x| is the Euclidean metric.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

Note that one can replace the Euclidean norm by any strictly convex convex norm in the previous argument. More generally, suppose that d1 and d2 are two projective metrics

• n Ω, then d = α1d1 + α2d2 is again a projective metric for any

α1, 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 convex domain... Note however that the above remark means that the projective metrics form a convex cone in the space of all functions Ω × Ω → R.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

It is often useful in studying metric spaces to also consider various generalization of the notion of distance function. The weakest notion is the following

• Definition. A weak metric on a set X is a function

δ : 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).

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

Here is a simple example: Let Ω be a convex subset of Rn and H ⊂ Rn be a hyperplane. The wall metric associated to H in Ω is the weak metric wH defined by wH(x, y) =

• 1

if H separates x from y,

• therwise.

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 for

the wall metric is equivalent to Pasch’s axiom applied to the plane containing those three points.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

The wall metric of a hyperplane is highly degenerate, but if we consider a large (finite) family of hyperplanes H1, H2, . . . , Hm ⊂ Rn, then these hyperplanes separate Ω in multiple 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

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

Note that the distance between two points in the same chambers is zero and the distance between two arbitrary points depends only

• n 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.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

To obtain a non degenerate metric from this procedure, one needs to replace the previous finite sum with an integral. We thus consider metrics of the type d(x, y) =

• H

wH(x, y)dµ(H) (2) Where H is the manifold of all hyperplanes in Rn and µ is a Borel measure on H. Note that formula (2) can simply be rewritten as d(x, y) = µ({H ∈ H | H separates x from y})

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

Theorem [H. Busemann] Let Ω be a convex domain in R2 and µ be a (non negative) Radon measure 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] = ∅}) > 0 for any non degenerate segment [x, y] ⊂ Ω. Then the formula d(x, y) = µ({H ∈ HΩ | H ∩ [x, y] = ∅}) (3) defines a non degenerate projective metric that is continuous with respect to the topology of R2.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

Furtermore, we have the following Proposition 1. [Crofton Formula.] The length of any rectifiable curve C in Ω for the metric d is given by ℓ(C) = µ({H ∈ HΩ | H ∩ C = ∅}) Proposition 2. If Ω = Rn and the measure µ is invariant under the action on H of the group of isometries of Rn, then the metric d is a multiple of the Euclidean metric. Likewise if Ω = Bn is the unit ball and the measure µ is invariant under the action on HBn of the group of projective tranformations

• f Bn, then the metric d is a multiple of the Hyperbolic metric.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

Let us examine under which condition on the measure µ the metric is strictly projective. Let us denote by H[x,y] the set of hyperplanes metting the segment [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])

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

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 is strictly projective if and only if for any triple x, y, z of non aligned points, the measure of the set of hyperplanes crossing both [x, z] and [z, y] is positive: µ(H[x,z] ∩ H[z,y]) > 0.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

In the converse direction we have the following result: Theorem Let d be a continuous projective metric in a convex domain Ω of the 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] = ∅})

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

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 for all non aligned triples x, y, z in Ω the set Ux,y,z = H[x,z] ∩ H[z,y]

• f lines crossing both [x, z] and [z, y]. For those sets he then

defines 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 set function is additive. We then extends this set function to any polygonal region and then, by standard measure theoretical arguments µ defines a Borel regular measure on Ω.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

The theorem still holds in higher dimension, but the construction produces 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.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

A quick view on the Finslerian viewpoint

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

If the measure µ is sufficiently regular, then the metric d is

• Finslerian. Let us quote two results in this direction.

Theorem (S. Ivanov 2009) If an arbitrary metric d is the plane is Lipschitz compared to the Euclidean metric, then it is a weak Finslerian metric: There exists an upper semicontinuous function F : Ω × Rn → R such that F(x, ·) is a norm for any x ∈ Ω and the distance d(x, y) is the infimum of the length of all curves γ joining them, where the length is defined as ℓ(γ) = t1

t0

F(γ(t), ˙ γ(t)dt

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

In dimension 2 we have Theorem (Pogorelov) Let d be a metric in a convex domain Ω ⊂ R2 be obtained from Busemann’s construction from a smooth measure dµ = fdA, then is Finslerian and the Finsler function F can be explicitely constructed from the function f . Some 10 years ago, Juan Carlos Alvarez Paiva reinterpreted Pogorelov’s result in the context of symplectic geometry and applied the method to the analysis of Finsler metrics with prescribed geodesics.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

We conclude with what is perhaps the main result on the Finslerian theory of projectively flat metrics: Theorem (Funk-Berwald, 1930’s) Let Ω be a bounded smooth and strongly convex domain. Then the Hilbert metric is the unique complete projective metric of constant flag curvature −1.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

• J. C. ´

Alvarez Paiva, Symplectic geometry and Hilbert’s fourth

• problem. 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 paths

between 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, 1979

European Math. Soc Pblishing House, 2014. Zhongmin Shen, Projectively flat Finsler metrics of constant flag curvature. Trans. Amer. Math. (2003).

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction

Statement of the IVth problem Historical context

Early results Busemann’s construction The Finsler Viewpoint

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