From Funk to Hilbert geometry

Transcription

1 Chapter 2 From Funk to Hilbert geometry Athanase Papadopoulos and Marc Troyanov Contents 1 Introduction 33 2 The Funk metric 35 3 The reverse Funk metric 41 4 Examples 42 5 The geometry of balls in the Funk metric 43 6 On the topology of the Funk metric 46 7 The triangle inequality and geodesics On the triangle inequality Geodesics and convexity in Funk geometry 51 8 Nearest points in Funk geometry 53 9 The infinitesimal Funk distance Isometries A projective viewpoint on Funk geometry Hilbert geometry Related questions 60 A Menelaus Theorem 61 B The classical proof of the triangle inequality for the Funk metric 64 References 66 1 Introduction The Funk metric associated to an open convex subset of a Euclidean space is a weak metric in the sense that it does not satisfy all the axioms of a metric: it is not symmetric, and we shall also allow the distance between two points to be zero; see Chapter 1 in this volume [19], where such metrics are introduced Weak metrics often occur in the calculus of variations and in Finsler geometry and the study of such metrics has been revived recently in low-dimensional topology and geometry by Thurston who introduced an asymmetric metric on Teichmüller space, which became the subject of intense research In this chapter, we shall sometimes use the expression metric instead of weak metric in order to simplify

2 34 Athanase Papadopoulos and Marc Troyanov The Funk weak metric F x; y/ is associated to an open convex subset of a Euclidean space R n It is an important metric for the subject treated in this Handbook, because the Hilbert metric H of the convex set is the arithmetic symmetrization of its Funk metric More precisely, for any x and y in, wehave H x; y/ D 1 2 F x; y/ C F y; x// : Independently of its relation with the Hilbert metric, the Funk metric is a nice example of a weak metric, and there are many natural questions for a given convex subset of R n that one can ask and solve for such a metric, regarding its geodesics, its balls, its isometries, its boundary structure, and so on Of course, the answer depends on the shape of the convex set, and it is an interesting aspect of the theory to study the influence of the properties of the (its degree of smoothness, the fact that it is a polyhedron, a strictly convex hypersurface, etc) on the Funk geometry of this set The same questions can be asked for the Hilbert geometry, and they are addressed in several chapters of this volume The Funk geometry is studied in Busemann s book The geometry of geodesics [2], although the name Funk metric is not used there, but it is used by Busemann in his later papers and books, see eg [3], and in the memoir [28] by Zaustinsky (who was a student of Busemann) We studied some aspects of this metric in [17], following Busemann s ideas But a systematic study of this metric is something which seems to be still missing in the literature, and the aim of this chapter is somehow to fill this gap Euclidean segments in are geodesics for the Funk metric, and in the case where the domain is strictly convex (that is, if there is no nonempty open Euclidean segment the Euclidean segments are the unique geodesic segments For a metric d on a subset of Euclidean space, the property of having the Euclidean segments d- geodesics is the subject of Hilbert s Problem IV (see Chapter 15 in this volume [16]) One version of this problem asks for a characterization of non-symmetric metrics on subsets of R n for which the Euclidean segments are geodesics The Funk metric is also a basic example of a Finsler structure, perhaps one of the most basic In the paper [17], we introduced the notions of tautological and of reversible tautological Finsler structures of the domain The Funk metric F is the length metric induced by the tautological weak Finsler structure, and the Hilbert metric H is the length metric induced by the reversible tautological Finsler structure of the domain This is in fact how Funk introduced his metric in 1929, see [9], [24] The reversible Finsler structure is obtained by the process of harmonic symmetrization at the level of the convex sets in the tangent spaces that define these structures, cf [18] This gives another relation between the Hilbert and the Funk metrics The Finsler geometry of the Funk metric is studied in some detail in Chapter 3 [24] of this volume A useful variational description of the Funk metric is studied in [27] There are also interesting non-euclidean versions of Funk geometry, see Chapter 13 in this volume ([20])

3 Chapter 2 From Funk to Hilbert geometry 35 In the present chapter, we start by recalling the definition of the Funk metric, and we give several basic properties of this metric, some of which are new, or at least formulated in a new way In Section 3, we introduce what we call the reverse Funk metric, that is, the metric r F defined by r F x; y/ D Fy;x/ This metric is much less studied than the Funk metric In Section 4, we give the formula for the Funk metric for the two main examples, namely, the cases of convex polytopes and the Euclidean unit ball In Section 5, we study the geometry of balls in the Funk metric Since the metric is non-symmetric, one has to distinguish between forward and backward balls We show that any forward ball is the image of by a Euclidean homothety This gives a rigidity property, namely, that the local geometry of the convex set determines the convex set up to a scalar factor In Section 6, we show that the topologies defined by the Funk and the reverse Funk metrics coincide with the Euclidean topology In Section 7, we give a proof of the triangle inequality for the Funk metric and at the same time we study its geodesics and its convexity properties In Section 8, we study the property of the nearest point projections of a point in on a convex subset of equipped with the Funk metric, and the perpendicularity properties In the case where is strictly convex, the nearest point projection is unique There is a formulation of perpendicularity to hyperplanes in in terms of properties of support hyperplanes for In Section 9, we study the infinitesimal (Finsler) structure associated to a Funk metric In Section 10, we study the isometries of a Funk metric In the case where is bounded and strictly convex, its isometry group coincides with the subgroup of affine transformations of R n that leave invariant In Section 11, we propose a projective viewpoint and a generalization of the Funk metric and in Section 12, we present some basic properties of the Hilbert metric which are direct consequences of the fact that it is a symmetrization of the Funk metric In the last section we give some new perspectives on the Funk metric, some of which are treated in other chapters of this volume Appendix A contains two classical theorems of Euclidean geometry (the theorems of Menelaus and Ceva) Menelaus Theorem is used in Appendix B to give the classical proof of the triangle inequality for the Funk metric 2 The Funk metric In this section, is a proper convex domain in R n, that is, is convex, open, nonempty, and R n We denote by x the closure of in R n and D x n the topological boundary of In the case of an unbounded domain, it will be convenient to add points at infinity to the To do so, we consider R n as an affine space in the projective space RP n and we denote by H 1 D RP n n R n the hyperplane at infinity We then denote by z the closure of in RP n and by D z n Observe D n H 1 For any two points x y in R n, we denote by Œx; y the closed affine segment joining them We also denote by Rx; y/ the affine ray starting at x and passing

4 36 Athanase Papadopoulos and Marc Troyanov through y and by zrx; y/ its closure in RP n Finally we set a x; y/ D zrx; y/ \ 2 RP n : We now define the Funk metric y x a Definition 21 (The Funk metric) The Funk metric on, denoted by F, is defined for x and y in,byf x; x/ D 0 and by jx aj F x; y/ D log jy aj if x y, where a D a x; y/ 2 RP n Here jp qj is the Euclidean distance between the points q and p in R n It is understood in this formula that if a 2 H 1, then F x; y/ D 0 This is consistent with the convention 1=1 D1 Let us begin with a few basic properties of the Funk metric Proposition 22 The Funk metric in a convex domain R n satisfies the following properties: (a) F x; y/ 0 and F x; x/ D 0 for all x;y 2 (b) F x; z/ F x; y/ C F y; z/ for all x;y;z 2 (c) F is projective, that is, F x; z/ D F x; y/ C F y; z/ whenever z is a point on the affine segment Œx; y (d) The weak metric F is non symmetric, that is, F x; y/ F y; x/ in general (e) The weak metric F is separating, that is, x y H) F x;y/>0, if and only if the domain is bounded (f) The weak metric F is unbounded Property (a) and (b) say that F is a weak metric Proof Property (a) follows from the fact that y 2 Œx; a x; y/, therefore jx a x; y/j jy a x; y/j 1

5 Chapter 2 From Funk to Hilbert geometry 37 and we have equality if y D x The triangle inequality (b) is not completely obvious The classical proof by Hilbert (who wrote it for the Hilbert metric) is given in Appendix B and a new proof is given in Section 7 To prove (c), observe that if y 2 Œx; z and x y z, then a x; y/ D a x; z/ D a y; z/ Denoting this common point by a, wehave jx aj jy aj jx aj F x; y/ C F y; z/ D log C log D log jy aj jz aj jz aj D F x; z/: Property (d) is easy to check, see Section 3 for more details Property (e) follows immediately from the definition and the fact that a convex domain in R n is unbounded if and only if it contains a ray, see [19], Remark 313 To prove (f), we recall that we always assume R n, and ; Let x be a point in and a a point and consider the open Euclidean segment x; a/ contained in For any sequence x n in this segment converging to a (with respect to the Euclidean metric), we have F x; x n / D log jx aj jx n!1as n!1 aj It is sometimes useful to see the Funk metric from other viewpoints If x, y and z are three aligned points in R n with z y, then there is a unique 2 R such that x D z C y z/ We call this number the division ratio or affine ratio of x with respect to z and y and denote it suggestively by D zx=zy Note also that >1 if and only if y 2 Œx; z We extend the notion of affine ratio to the case z 2 H 1 by setting zx=zy D 1 if z 2 H 1 We then have F x; y/ D log/; where is the affine ratio of x with respect to a D a x; y/ and y Proposition 23 Let x and y be two points in the convex domain and set a D a x; y/ Let hw R n! R be an arbitrary linear form such that hy/ hx/ Then ha/ hx/ F x; y/ D log : ha/ hy/ Proof We first observe that a linear function hw R n! R is either constant on a given ray Rx; y/, or it is injective on that ray If a D a x; y/ 2 H 1, then ha/ D lim hx C ty x// D hx/ C thy/ hx// D 1; t!1 and the proposition is true since log1=1/ D log1/ D 0 D F x; y/ Ifa 62 H 1, then the division ratio D ax=ay > 1 and we have a x/ D a y/ and F x; y/ D log/ Since hw R n! R is linear, then ha/ hx// D ha/ hy// and the proposition follows at once We now recall some important notions from convex geometry For these and other classical results on convex geometry, the reader can consult the books by Eggleston [7], Fenchel [8], Valentine [26] and Rockafellar [22]

6 38 Athanase Papadopoulos and Marc Troyanov Definition 24 Let R n be a proper convex domain and a be a (finite) boundary point We assume that contains the origin 0 Asupporting functional at a for is a linear function hw R n! R such that ha/ D 1 and hx/ < 1 for any point x in 1 The hyperplane H Dfz j hz/ D 1g is said to be a support hyperplane at a for If is unbounded, then the hyperplane at infinity H 1 is also considered to be a support hyperplane A basic fact is that any boundary point of a convex domain admits one or several supporting functional(s) Let us denote by S the set of all supporting functionals of Then p x/ D supfhx/ j h 2 S g is the Minkowski functional of This is the unique weak Minkowski norm such that Dfx 2 R n j p x/<1g: We have the following consequences of the previous proposition: Corollary 25 Let R n be a convex domain containing the origin and let h be a supporting functional for We then have 1 hx/ F x; y/ log 1 hy/ for any x;y 2 Furthermore we have equality if and only if either a D a x; y/ 2 H 1 and hx/ D hy/,ora 62 H 1 and ha/ D 1 Proof If hx/ D hy/ there is nothing to prove, we thus assume that hx/ hy/ Note that this condition means that the line through x and y is not parallel to the supporting hyperplane H Dfh D 1g We first assume that a D a x; y/ 62 H 1, The hypothesis hx/ hy/ implies ha/ hy/ Because the function a 7! log jx aj is strictly monotone decreasing jy aj and ha/ 1, we have by Proposition 23: F x; y/ D log ha/ hx/ ha/ hy/ log 1 hx/ : 1 hy/ The equality holds if and only if ha/ D 1 Suppose now that a 2 H 1 ; this means that F x; y/ D 0 and the ray Rx; y/ is contained in In particular, we have hx/ C hy x/ D hx C y x//<1 for all 0; which implies hy/ hx/ D hy x/ 0 Since we assumed hx/ hy/, we have hy/ < hx/ and therefore F x; y/ D 0>log 1 hx/ : 1 hy/ 1 The reader should not mistake the notion of supporting functional for that of support function h W R n! R defined as h x/ D supfhx;yi jy 2 g

7 Chapter 2 From Funk to Hilbert geometry 39 Corollary 26 If R n is a convex domain containing the origin, then ± F x; y/ D max 0; sup h2s log : 1 hx/ 1 hy/ Proof We first assume that F x;y/>0 Then a D a x; y/ 62 H 1 Let us denote by ˆx; y/ the right hand side in the above formula Then Proposition 23 implies that F x; y/ ˆx; y/ and Corollary 25 implies the converse inequality If F x; y/ D 0, then Rx;y/<0 For any supporting function h, we then have hx C ty x// D hx/ C thy/ hx// < 1 for any t > 0 This implies that hy/ hx/ and it follows that ˆx; y/ D 0 Notice that for bounded domains, the previous formula reduces to 1 hx/ F x; y/ D sup log : h2s 1 hy/ This can also be reformulated as follows (compare to Theorem 1 of [27]): Corollary 27 The Funk metric in a bounded convex domain R n is given by distx; H / F x; y/ D sup log ; H disty; H / where the supremum is taken over the set of all support hyperplanes H for and distx; H / is the Euclidean distance from x to H The next consequence of Proposition 23 is the following relation between the division ratio of three aligned points in a convex domain and the Funk distances between them Corollary 28 Let x, y and z be three aligned points in the convex domain R n Suppose that F x;y/>0and z D x C ty x/ for some t 0 Then t D efx;y/ e Fx;z/ 1/ e F x;z/ e F x;y/ 1/ ; (21) F x; z/ D F x; y/ log e Fx;y/ C t 1 e Fx;y/ / : (22) Proof Choose a supporting functional h for at the point a D a x; y/ D a x; z/ Then we have, from Corollary 25: e F x;y/ D Therefore e Fx;z/ 1 D e F x;z/ 1 hx/ 1 hy/ and e F x;z/ D 1 hz/ 1 hx/ 1 hx/ 1 hz/ 1 D 1 hx/ 1 hz/ : hz/ hx/ ; 1 hx/

8 40 Athanase Papadopoulos and Marc Troyanov and likewise e F x;y/ 1 e F x;y/ D hy/ hx/ : 1 hx/ Thus e Fx;y/ e Fx;z/ 1/ hz/ hx/ hz x/ D D e F x;z/ e F x;y/ 1/ hy/ hx/ hy x/ D t: This proves Equation (21) To prove the equation (22), we now resolve for e F x;z/ This gives us e F x;z/ 1 e F x;z/ e F x;z/ D which is equivalent to (22) D t ef x;y/ 1 e F x;y/ e F x;y/ e F x;y/ C t 1 e F x;y/ ; It is useful to observe that computing the Funk distance between two points in is a one-dimensional operation More precisely, if S D Œa 1 ;a 2 R n is a compact segment in R n containing the points x and y in its interior with y 2 Œx; a 2, we shall write F S x; y/ D log jx a 2j jy a 2 j : Although S is not an open set, F S x; y/ clearly corresponds to the one-dimensional Funk metric in the relative interior of S Proposition 29 The Funk distance between two points x and y in is given by F x; y/ D inf ff S x; y/ j S is a segment in containing x and yg : Proof We identify S with a segment in R with b<xy<aand observe that the function a 7! log jx aj is strictly monotone decreasing jy aj This result makes an analogy between the Funk metric and the Kobayashi metric in complex geometry, see [13] It has the following immediate consequences: Corollary 210 (i) If 1 2 are convex subsets of R n, then F 1 F 2 with equality if and only if 1 D 2 (ii) Let 1 and 2 be two open convex subsets of R n Then, for every x and y in 1 \ 2, we have F 1 \ 2 x; y/ D maxf 1 x; y/; F 2 x; y//: (iii) Let be a nonempty open convex subset of R n, let 0 be the intersection of with an affine subspace of R n, and suppose that 0 ; Then, F 0 is the metric induced by F on 0 as a subspace of ; F /

9 Chapter 2 From Funk to Hilbert geometry 41 3 The reverse Funk metric Definition 31 The reverse Funk metric in a proper convex domain is defined as jy bj r F x; y/ D F y; x/ D log ; jx bj where b D a y; x/ y x b Figure 1 The reverse Funk metric Proposition 32 The reverse Funk metric in a convex domain R n is a projective weak metric It is unbounded and non-symmetric and it is separating if and only if the domain is bounded The proof is a direct consequence of Proposition 22 An important difference between the Funk metric and the reverse Funk metric is the following: Proposition 33 Let be a bounded convex domain in R n and x be a point in Then the function y! r F x; y/ is bounded Proof Define x and ı by x D inf b2@ jx bj; and ı D sup a;b2@ ja bj: Observe that ı is the Euclidean diameter of, thus ı<1since is bounded We also have x >0 The proposition follows from the inequality : r F x; y/ log ı x In particular the reverse Funk metric r F is not bi-lipschitz equivalent to the Funk metric F

10 42 Athanase Papadopoulos and Marc Troyanov 4 Examples Example 41 (Polytopes) An (open) convex polytope in R n is defined to be an intersection of finitely many half-spaces: Dfx 2 R n j j x/<s j ;1 j kg; where j W R n! R is a nontrivial linear form for all j The Funk distance between two points in such a polytope is given by ± sj F x; y/ D max 0; max 1j k log j x/ s j j : y/ The proof is similar to that of Corollary 26 As a special case, let us mention that the Funk metric in R n C is given by F R n C x; y/ D max 1in max 0; log x i y i ±: Observe that the map x D x i / 7! u D u i /, where u i D logx i /, is an isometry from the space R n C ;F R n C / to Rn with the weak Minkowski distance ıu; v/ D max 1in max ¹0; u i v i º: Example 42 (The Euclidean unit ball) The following is a formula for the Funk metric in the Euclidean unit ball B R n : p! jy xj2 jx ^ yj F B x; y/ D log 2 Cjxj 2 hx;yi p ; (41) jy xj2 jx ^ yj 2 jyj 2 Chx;yi where jx ^ yj D p jxj 2 jyj 2 hx;yi 2 is the area of the parallelogram with sides! 0x,! 0y x y a a 2 0 a 1 Proof If x D y, there is nothing to prove, so we assume that x y Let us set a D a B x; y/ D Rx; y/ Using Proposition 23 with the linear form hz/ Dhy x;zi we get hy x;ai hy x;xi hy x;aicjxj 2 hx;yi F B x; y/ D log D log : hy x;ai hy x;yi hy x;ai jyj 2 Chx;yi

11 Chapter 2 From Funk to Hilbert geometry 43 So we just need to compute hy x;ai This is an exercise in elementary Euclidean geometry Let us set u D y x jy xj and a 1 Dhu; aiu; a 2 D a a 1 : Then a D a 1 C a 2 and a 1 is a multiple of y x while a 2 is the orthogonal projection of the origin O of R n on the line through x and y In particular the altitude of the triangle Oxy is equal to ja 2 j, therefore AreaOxy/ D 1 2 jx ^ yj D1 2 ja 2jjy xj: Observe now that hu; ai >0and jaj 2 Dja 1 j 2 Cja 2 j 2 D 1, we thus have hy x;ai 2 Djy xj 2 hu; ai 2 The desired formula follows immediately Djy xj 2 ja 1 j 2 Djy xj 2 1 ja 2 j 2 / Djy xj 2 jx ^ yj 2 : 5 The geometry of balls in the Funk metric Since we are dealing with non-symmetric distances, we need to distinguish between forward and backward balls For a point x in and >0,weset B C x; / Dfy 2 B j F x;y/<g (51) and we call it the forward open ball (or right open ball) centered at x of radius In a symmetric way, we set B x; / Dfy 2 B j F y;x/<g (52) and we call it the backward open ball (also called the left open ball) centered at x of radius Note that the open backward balls of the Funk metric are the open forward balls of the reverse Funk metric, and vice versa We define closed forward and closed backward balls by replacing the inequalities in (51) and (52) by non strict inequalities, and in the same way we define forward and backward spheres by replacing the inequalities by equalities In Funk geometry, the backward and forward balls have in general quite different shapes and different properties Proposition 51 Let be a proper convex open subset of R n equipped with its Funk metric F, let x be a point in and let be a nonnegative real number We have:

12 44 Athanase Papadopoulos and Marc Troyanov The forward open ball B C x; / is the image of by the Euclidean homothety of center x and dilation factor 1 e / The backward open ball B x; / is the intersection of with the image of by the Euclidean homothety of center x and dilation factor e 1/, followed by the Euclidean central symmetry centered at x A forward and a backward ball in Funk geometry The forward ball is always relatively compact in, while the closure of the backward ball may meet the if its radius is large enough Proof Let y x be a point in IfF x; y/ D 0, then the ray Rx; y/ is contained in, and for any z on that ray, we have F x; z/ D 0 Therefore the ray is also contained in Bx; / IfF x; y/ D 0, then a D a x; y/ 62 H 1 and we have the following equivalent conditions for any point y on the segment Œx; a : y 2 B C jx aj x; / () log jy aj < () jx aj <e jy aj De jx aj jy xj/ () jy xj <1 e /jx aj: This proves the first statement The proof of the second statement is similar; let us set b D a y; x/, then for any point y 2 Œx; a we have x 2 Œb; y, therefore y 2 B jy bj x; / () log jx bj < () e jx bj > jy bj Djy xjcjx bj () e 1/jx bj > jy xj: Thus, for instance, if is the interior a Euclidean ball in R n, then any forward ball for the Funk metric B C x 0 ;ı/ is also a Euclidean ball However, its Euclidean center is not the center for the Funk metric (unless x 0 is the center of ) Considering Example 42, if is the Euclidean unit ball and B C x 0 ;/ is the Funk ball of radius and center x 0 in, then y 2 B C x 0 ;/if and only if F x 0 ;y/ Using

13 Chapter 2 From Funk to Hilbert geometry 45 Formula (41), we compute that this is equivalent to kyk 2 2e hy; x 0 ice 2 kx 0 k 2 1 e / 2 : This set describes a Euclidean ball with center z 0 D e x 0 and Euclidean radius r D 1 e / We deduce the following local-implies-global property of Funk metrics The meaning of the statement is clear, and it follows directly from Proposition 51 Corollary 52 We can reconstruct the of from the local geometry at any point of Corollary 53 For any points x and x 0 in a convex domain equipped with its Funk metric and for any two positive real numbers ı and ı 0, the forward balls B C x; ı/ and B C x 0 ;ı 0 / are either homothetic or a translation of each other Proof This follows from Proposition 51 and the fact that the set of Euclidean transformations which are either homotheties or translations form a group (sometimes called the the group of dilations, see eg [6]) Remark 54 The previous corollary also holds for backward balls B x; ı/ of small enough radii Remarks 55 In the case where the convex set is unbounded, the forward and backward open balls of its Funk metric are always noncompact If is bounded then for any x 2 and for large enough we have B x; / D This follows from Proposition 33 In particular, the closed backwards balls are not compact for large radii The forward open balls are geodesically convex if and only if is strictly convex Remark 56 The property for a weak metric on a subset of R n to have all the right spheres homothetic is also shared by the Minkowski weak metrics on R n Indeed, it is easy to see that in a Minkowski weak metric, any two right open balls are homothetic (Any two right spheres of the same radius are translates of each other, and it is easy to see from the definition that any two spheres centered at the same point are homothetic, the center of the homothety being the center of the balls) Thus, Minkowski weak metrics share with the Funk weak metrics the property stated in Proposition 51 Busemann proved that in the setting of Desarguesian spaces, these are the only examples of spaces satisfying this property (see the definition of a Desarguesian space and the statement of this result in Section 6 of Chapter 1 in this volume ([19])) We state this as the following: Theorem 57 (Busemann [3]) A Desarguesian space in which all the right spheres of positive radius around any point are homothetic is either a Funk space or a Minkowski space

14 46 Athanase Papadopoulos and Marc Troyanov 6 On the topology of the Funk metric Proposition 61 The topology induced by the Funk or reverse Funk metric in a bounded convex domain in R n coincides with the Euclidean topology in that domain Proof The proof consists in comparing the balls in the Euclidean and the Funk (or reverse Funk) geometries Let us fix a point x in Then there exists 0< x ƒ x < 1 such that for any we have x j xj ƒ x : If we denote by B C x; / the forward ball with center x and radius in the Funk metric, then Proposition 51 implies that y C x; / H) 1 e / x jy xj 1 e /ƒ x : In other words, if E Bx; ı/ denotes the Euclidean ball with center x and radius ı, then E Bx; 1 e / x / B C x; / E Bx; 1 e /ƒ x /: (61) This implies that the families of balls B C x; / and E Bx; ı/ are sub-bases for the same topology For the backward balls B x; /, the second part of Proposition 51 implies the following y x; / H) e 1/ x jy xj e 1/ƒ x ; provided e 1/ 1 This implies that for log2/ we have E Bx; e 1/ x / B x; / E Bx; e 1/ƒ x /; and therefore the family of backward balls B x; / also generates the Euclidean topology For general convex domains, bounded or not, we have the following weaker result on the topology: Proposition 62 For any convex domain in R n, F is a continuous functions on Proof We first consider the case where is bounded Suppose first that x and y are distinct points in and let x n, y n be sequences in converging to x and y respectively Taking subsequences if necessary, we may assume that x n y n for all n Then a n D a x n ;y n / is well defined and this sequence converges to a D a x; y/

15 Chapter 2 From Funk to Hilbert geometry 47 Since a y we have lim F x n ;y n / D lim log jxn a n j n!1 n!1 jy n a n j jx aj D log jy aj D F x; y/: Assume now that x D y and let x n ;y n 2 be sequences converging to x such that x n y n for all n Wehave jxn a n j F x n ;y n / D log jy n a n j jyn a n / C x n y n /j D log jy n a n j log 1 C jx n y n j : jy n a n j Since y n 2 converges to a point x in, wehaveı D sup b2@ jy n bj 1 < 1 We then have F x n ;y n / log 1 C ıjx n y n j/! 0; since jx n y n j!0 If is unbounded, we set R D \ E Bx; R/ where E Bx; R/ is the Euclidean ball of radius R centered at the origin It is easy to check that F R converges uniformly to F on every compact subset of as R!1 The continuity of F follows therefore from the proof for bounded convex domains For the next result we need some more definitions: Definition 63 Let ı be a weak metric defined on a set X A sequence fx k g in X is forward bounded if sup ıx k ;x m /<1 where the supremum is taken over all pairs k, m satisfying m k Note that this definition corresponds to the usual notion in the case of a usual (symmetric) metric space We then say that the weak metric space X; ı/ is forward proper, orforward boundedly compact if every forward bounded sequence has a converging subsequence The sequence fx k g is forward Cauchy if lim k!1 sup ıx k ;x m / D 0: mk The weak metric space X; ı/ is forward complete if every forward Cauchy sequence converges We define backward properness and backward completeness in a similar way

16 48 Athanase Papadopoulos and Marc Troyanov Proposition 64 The Funk metric in a convex domain R n is forward proper (and in particular forward complete) if and only if is bounded The Funk metric is never backward complete Proof For a convex domain, the ball inclusions (61) immediately imply that forward complete balls are relatively compact; this implies forward properness If is unbounded, then it contains a ray and such a ray contains a divergent sequence fx k g such that F x k ;x m / D 0 for any m k therefore F is not complete To prove that the Funk metric is never backward complete, we consider an affine segment Œa; b 2 R n with a b and such that Œa; b Dfa; bg Set x k D b C 1 a b/ Ifm k, then k F x k ;x m / D log jx m aj jx k aj ; which converges to 0 as k!1 Since the sequence fx k g has no limit in, we conclude that F is not backward complete Remark 65 The previous proposition also says that in a bounded convex domain, the Funk metric is forward complete and the reverse Funk metric is not 7 The triangle inequality and geodesics In this section, we prove the triangle inequality for the Funk metric and give a necessary and sufficient condition for the equality case We also describe all the geodesics of this metric 71 On the triangle inequality Theorem 71 If x, y and z are three points in a proper convex domain, then the triangle inequality F x; y/ C F y; z/ F x; z/ (71) holds Furthermore we have equality F x; y/ C F y; z/ D F x; z/ if and only if the three points a x; y/; a y; z/; a x; z/ 2 (72) are aligned in RP n Before proving this theorem, let us first recall a few additional definitions from convex geometry: Let R n be a convex domain Then it is known that its closure x is also convex A convex subset D x is a face of x if for any x;y 2 D and any

17 Chapter 2 From Funk to Hilbert geometry 49 0<<1we have 1 /x C y 2 D H) Œx; y D: The empty set and x are also considered to be faces A face D x is called proper if D x and D 6D ; A face D is said to be exposed if there is a supporting hyperplane H for such that D D H \ x Recall that a support hyperplane is a hyperplane H that and H \ D; It is easy to prove that every proper face is contained in an exposed face In fact every maximal proper face is exposed A point x x is an exposed point of if fxg is an exposed face, that is, if there exists a hyperplane H R n such that H \ x Dfxg If is bounded, then x is the closure of the convex hull of its exposed points (Straszewicz s Theorem) A point x 2 x is an extreme point if x nfxg is still a convex set Such a point belongs to the and if is bounded, then x is the convex hull of its extreme points (Krein Milman s Theorem) Every exposed point is an extreme point, but the converse does not hold in general The following result immediately follows from the definitions: Lemma 72 The following are equivalent conditions for a convex domain R n : (i) Every boundary point is a extreme point (ii) Every boundary point is an exposed point (iii) The does not contain any non-trivial segment If one of these conditions holds, then is said to be strictly convex The following result will play an important role in the proof of Theorem 71: Lemma 73 Let be bounded convex domain and x, y, z three points in Then the following are equivalent: (a) There exists a proper face Q such that a x; y/; a y; z/; a x; z/ 2 D: (b) The three points a x; y/; a y; z/ and a x; z/ are aligned in RP n Proof Let us set a D a x; y/, b D a y; z/ and c D a x; z/ Ifx, y and z are aligned, then a D b D c Otherwise, a, b and c belong to the 2-plane containing x, y, z Therefore if a, b, c belong to a proper face D, then those three points are contained in the interval \ D This proves the implication (a) H) (b) The converse implication (b) H) (a) is obvious Proof of Theorem 71 We first consider F x; z/ D 0 In this case the inequality (71) is trivial and we have c 2 H 1 We then have equality in (71) if and only if F x; y/ D F y; z/ D 0, which is equivalent to a 2 H 1 and b 2 H 1 (the points a and b are as in the proof of the previous lemma) It then follows from Lemma 73 that a, b, c lie on some line (at infinity)

18 50 Athanase Papadopoulos and Marc Troyanov We now consider the case F x;z/>0, that is, c 62 H 1 Choose a supporting functional h at the point c (that is, hc/ D 1) We then have from Corollary 25: 1 hx/ F x; z/ D log 1 hz/ 1 hx/ 1 hy/ D log C log 1 hy/ 1 hz/ F x; y/ C F y; z/: This proves the triangle inequality Using again Corollary 25, we see that we have equality if and only if 1 hx/ F x; y/ D log 1 hy/ and 1 hy/ F y; z/ D log ; 1 hz/ and this holds if and only if one of the following cases holds: Case 1 We have a 62 H 1 and b 62 H 1 In that case, ha/ D hb/ D 1 D hc/ The three points a, b, c belong to the face D \fh D 1g and we conclude by Lemma 73 that a, b, c lie on some line Case 2 We have a 2 H 1 and b 62 H 1 In that case hb/ D 1 D hc/ and hx/ D hy/ This implies that the line through x and y is parallel to the hyperplane fh D 1g and therefore the point a 2 zrx; y/ \ H 1 belongs to the support hyperplane fh D 1g Since hb/ D 1, the three points a, b, c belong to the face D Q \fh D 1g and we conclude by Lemma 73 Case 3 We have a 62 H 1 and b 2 H 1 The argument is the same as in Case 2 To complete the proof, we need to discuss the case a 2 H 1 and b 2 H 1 In this case, we would have hx/ D hy/ and hy/ D hz/ and this is not possible Indeed, we have c D x C z x/ for some and the equality hz/ D hx/ would lead to the contradiction 1 D hc/ D hx C z x// D hx/ C hz/ hx// D hx/ < 1: We thus proved in all cases that the equality holds in (71) if and only if the points a, b and c are aligned in RP n Corollary 74 Let x and z be two points in a proper convex domain R n Suppose that a x; z/ is an exposed point Then for any point y 62 Œx; z we have F x;z/<f x; y/ C F y; z/

19 Chapter 2 From Funk to Hilbert geometry Geodesics and convexity in Funk geometry We now describe geodesics in Funk geometry Let us start with a few definitions Definitions 71 A path in a weak metric space X; d/ is a continuous map W I! X, where I is an interval of the real line The length of path W Œa; b! X is defined as NX 1 Length/ D sup dt i /; t ic1 //; id0 where the supremum is taken over all subdivisions a D t 0 <t 1 < <t N D b Note that in the case where the weak metric d is non-symmetric, the order of the arguments is important The path W Œa; b! X is a geodesic if da/; b// D Length/ The weak metric space X; d/ is said to be a weak geodesic metric space if there exists a geodesic path connecting any pair of points It is said to be uniquely geodesic if this geodesic path is unique up to reparametrization A subset A X is said to be geodesically convex if given any two points in A, any geodesic path joining them is contained in A Lemma 75 The path W Œa; b! X is geodesic if and only if for any t 1, t 2, t 3 in Œa; b satisfying t 1 t 2 t 3 we have dt 1 /; t 3 // D dt 1 /; t 2 // C dt 2 /; t 3 // The proof is an easy consequence of the definitions Let us now consider a proper convex domain R n and a proper face Q For any point p 2, we denote by C p D/ Dfv 2 R n j v D 0 or xrp; p C v/ \ D ;g: (73) Here xrp; p C v/ is the extended ray through p and p C v in RP n (Recall that the projective space RP n is considered here as a completion of the Euclidean space R n obtained by adding a hyperplane at infinity; the completion of the ray is then its topological completion) Observe that C p D/ is a cone in R n at the origin, its translate p C C p D/ is the cone over D with vertex at p We then have the following Theorem 76 Let W Œ0; 1! be a path in a proper convex domain of R n Then is a geodesic for the Funk metric in if and only if there exists a face Q such that for any t 1 <t 2 in Œ0; 1 we have t 2 / t 1 / 2 C t1 /D/: In particular if a x; y/ is an exposed point, then there exists a unique (up to reparametrization) geodesic joining x to y, and this geodesic is a parametrization of the affine segment Œx; y Proof This is a direct consequence of Theorem 71 together with Lemma 73 and Lemma 75

20 52 Athanase Papadopoulos and Marc Troyanov For smooth curves we have the following Corollary 77 Let W Œ0; 1! be a C 1 path in a proper convex domain of R n Then is a Funk geodesic if and only if there exists a face D such that Pt/ 2 C t/ D/ for any t 2 Œa; b D A typical smooth geodesic in Funk geometry: all tangents to the curve meet the same face For a subset of R n, equipped with a (weak) metric F, we have two notions of convexity: affine convexity, saying that for every pair of points in, the affine (or Euclidean) geodesic joining them is contained in, and geodesic convexity, saying that for every pair of points in, the F -geodesic joining them is contained in From the preceding results, we have the following consequence on geodesic convexity of subsets for the Funk metric Corollary 78 Let be a bounded convex domain of R n Then the following are equivalent: (1) is strictly convex (2) is uniquely geodesic for the Funk metric (3) A subset A is geodesically convex for the Funk metric if and only if A is affinely convex (4) The forward open balls in are geodesically convex with respect to the Funk metric F Proof (1) H) (2) immediately follows from Theorem 76 (2) H) (3) is obvious and (3) H) (4) follows from Proposition 51

21 Chapter 2 From Funk to Hilbert geometry 53 q x p y a b We now prove (4) H) (1) by contraposition Let us assume that is not strictly convex, so that there exists a non-trivial segment Œa; On can then find a supporting functional h for such that ha/ D hb/ D 1 Let us chose a segment Œx; y such that a x; y/ 2 Œa; b and another segment Œp; q such that p 2 Œx; y and hq/ D hp/ We also assume x p y and F p;q/>ıwd F p; x/ C F p; y/ Observe that we then have hx/ < hp/ D hq/ < hy/ If a x; q/ 2 Œa; b and a q; y/ 2 Œa; b, then the proof is finished since in this case F x; y/ D log hx/ hx/ D log hy/ hq/ hq/ hy/ D F x; q/ C F q; y/: Since x;y 2 B C p; ı/ while q 62 B C p; ı/, we conclude that the forward ball B C p; ı/ is not geodesically convex If a x; q/ 62 Œa; b or a q; y/ 62 Œa; b, we let c D a x; y/ and we consider the Euclidean homothety f W R n! R n centered at c with dilation factor <1 Let p 0 D f p/, q 0 D f q/; x 0 D f x/; y 0 D f y/ and q 0 D f q/ It is now clear that if >0is small enough, then a x 0 ;q 0 / 2 Œa; b and a q 0 ;y 0 / 2 Œa; b It is also clear that one can find a number ı 0 such that x 0 ;y 0 2 B C p 0 ;ı 0 / while q 0 62 B C p 0 ;ı 0 / The previous argument shows then that F x 0 ;y 0 / D F x 0 ;q 0 / C F q 0 ;y 0 / and therefore B C p 0 ;ı 0 / is not geodesically convex Remark 79 Note the formal analogy between Corollary 78 and the corresponding result concerning the geodesic segments of a Minkowski metric on R n : if the unit ball of a Minkowski metric is strictly convex, then the only geodesic segments of this metric are the affine segments 8 Nearest points in Funk geometry Let R n be a convex set equipped with its Funk metric F Definition 81 Let x be a point in and let A be a subset of A point y in A is said to be a nearest point,orafoot (in Buseman s terminology), for x on A if F x; y/ D F x; A/ WD inf z2a F x; z/:

22 54 Athanase Papadopoulos and Marc Troyanov It is clear from the continuity of the function y 7! F x; y/ that for any closed non-empty subset A and any x 2, there exists a nearest point y 2 A This point need not be unique in general Proposition 82 For a proper convex domain R n, the following properties are equivalent: a) is strictly convex b) For any closed convex subset A and for any x 2, there is a unique nearest point y 2 A Proof let x be a point in and assume r D F x;a/>0 Suppose that y and z are two nearest points of A for x For any point w on the segment Œy; z we have F x; w/ r because the closed ball xb C x; r/ is convex Since A is also assumed to be convex, we have w 2 A and therefore F x; w/ r We conclude that F x; w/ D r, that is, w xb C x; r/ From Proposition 51, we know that if is strictly convex, then xb C x; r/ is also strictly convex and we conclude that y D z It follows that we have a unique nearest point on A for x This proves (a) H) (b) To prove (b) H) (a), we assume by contraposition that is strictly convex Again from Proposition 51, we know that the forward ball xb C x; r/ is not strictly convex In C x; r/ contains a non trivial segment A D Œy; z Any point in the convex set A is a nearest point to x and this completes the proof Proposition 83 Let A be an affinely convex closed subset of a proper convex domain R n and let x 2 n A A point y 2 A is a nearest point in A for x if and only if either F x; y/ D 0 or there exists a hyperplane R n which contains y, which separates A and x and which is parallel to a support hyperplane H for at a D a x; y/ Proof We assume F x;y/>0(otherwise, there is nothing to prove) First, suppose there exists a hyperplane R n containing y and separating A from x and which is parallel to a support hyperplane H for at a Let h be the corresponding supporting functional Then we have, from our hypothesis, hx/ < hy/ D inf hz/: z2a From Proposition 23 and Corollary 25 we then have 1 hx/ 1 hx/ F x; y/ D log D inf 1 hy/ log F x; A/; z2a 1 hz/ therefore y is a nearest point on A for x To prove the converse, we assume that y is a nearest point on A for x Set r D F x; y/ D F x; A/, then, by definition, the forward open ball B C x; r/ and the set A are disjoint Since both sets are affinely convex, there exists a hyperplane that separates them Note that is then a support hyperplane at y for the ball B C x; r/

23 Chapter 2 From Funk to Hilbert geometry 55 We conclude from Proposition 51 that is parallel to a support hyperplane H for at a There are several possible notions of perpendicularity in metric spaces The following definition is due to Busemann (see [2], page 103) Definition 84 (Perpendicularity) Let A be a subset of and p a point in A A geodesic W I! is said to be perpendicular to A at p if the following two properties hold: (1) p D t 0 / for some t 0 2 I, (2) for every t 2 I, p is a nearest point for t/ on A From the previous results we have the following Corollary 85 Let x be a point in a convex domain and a be a boundary point If R n is a hyperplane containing x, then the ray Œx; a/ is perpendicular to \ if and only if is parallel to a support hyperplane H a of at a If b is another boundary point, then the line a; b/ is perpendicular to \ if and only if \ a; b/ ;and is parallel to both a support hyperplane H a at a and a support hyperplane H b at b 9 The infinitesimal Funk distance In this section, we consider a convex domain R n and a point p in We define a weak distance ˆp D ˆ;p on R n as the limit ˆpx; y/ D lim t&0 F ;p p C tx;p C ty/ : t Theorem 91 The weak metric ˆp at a point p in R n is a Minkowski weak metric in R n Its unit ball is the translated domain p D p Proof Choose a supporting functional h for and set 1 hp C tx/ ' h t/ D log D log 1 C 1 hp C ty/ thy x/ 1 hp/ thy/ The first two derivatives of this functions are ' 0 h t/ D 1 hp// hy x/ 1 hp/ thx//1 hp/ thy// ; and ' 00 1 hp//hx/ C hy// 2t hx/hy//1 hp// hy x/ h t/ D 1 hp/ thx// 2 1 hp/ thy// 2 : :

24 56 Athanase Papadopoulos and Marc Troyanov We have in particular ' 0 hy x/ h 0/ D 1 hp/ ; and the second derivative is uniformly bounded in some neighborhood of p More precisely, given a relatively compact neighborhood U xu of p, one can find a constant C which depends on U but not on h such that j' 00 h t/j C; for any x;y 2 U and jtj 1and any support function h We have, from Taylor s formula, hy x/ ' h t/ D t 1 hp/ C t 2 x; y; h/; where jx; y; h/j C Using Corollary 26 we have thy x/ F ;p p C tx;p C ty/ D sup ' h t/ D sup h h 1 hp/ C Ot2 /; where the supremum is taken over the set S of all support functions for Therefore hy x/ ˆpx; y/ D sup h2s 1 hp/ : We then see that ˆpx; y/ is weak Minkowski distance (see Chapter 1 in this Handbook) hy/ ˆp0; y/ 1 () sup h2s 1 hp/ 1 () hy/ 1 hp/ for all support functions h of () hp C y/ 1 for all h () y 2 x p; this means that the unit ball of ˆp is the translate of by p Remark 92 The Funk metric of a convex domain is in fact Finslerian, and the previous theorem means that the Finslerian unit ball at any point p coincides with the domain itself with the point p as its center This is why the Funk metric was termed tautological in [17] The Finslerian approach to Funk geometry is developed in [24] 10 Isometries It is clear from its definition that the Funk metric is invariant under affine transformation Conversely, we have the following:

25 Chapter 2 From Funk to Hilbert geometry 57 Proposition 101 Let 1 and 2 be two bounded convex domains in R n Assume that there exists a Funk isometry f W U 1! U 2, where U i is an open convex subset of i, i D 1; 2 If 2 is strictly convex, then f is the restriction of a global affine map of R n that maps 1 to 2 Proof Let x and y be two distinct points in U 1 Then, for any z 2 Œx; y, wehave F 2 f x/; f y// F 2 f x/; f z// F 2 f z/; f y// D F 1 x; y/ F 1 x; z/ F 1 z; y/ D 0: Since 2 is assumed to be strictly convex, this implies that fz/belongs to the line through fx/and fy/ We can thus define the real numbers t and s by z D x C ty x/ and fz/d fx/c sfy/ f x//: It now follows from Corollary 28 and the fact that f is an isometry that s D t We thus have proved that for any x;y 2 U 1 and any t 2 Œ0; 1 we have fxc ty x// D fx/c tfy/ f x//: (101) This relation easily implies that f is the restriction of a global affine mapping We immediately deduce the following: Corollary 102 The group of Funk isometries of a strictly convex bounded domains R n coincides with the subgroup of the affine group of R n leaving invariant Remark 103 The conclusion of the corollary may fail for unbounded domains For instance, if is the upper half plane fx 2 >0g in R 2, then F x; y/ D maxf0; logx 2 =y 2 /g and any map f W! of the type fx 1 ;x 2 / D ax 1 C b; x 2 /, where a 0 and W R! R is arbitrary, is an isometry 11 A projective viewpoint on Funk geometry In this section we consider the following generalization of Funk geometry: We say that a subset U RP n is convex if it does not contain any full projective line and if the intersection of any projective line L RP n with U is a connected set If U and are connected domains in RP n with U, and if x, y are two distinct points in We denote by a x; y/ and! x; y/ the boundary points on the line L through x and y appearing in the order a, y, x,!

26 58 Athanase Papadopoulos and Marc Troyanov Definition 111 The relative Funk metric for U is defined by F ;U x; x/ D 0 and by jy!j jx aj F ;U x; y/ D log ; jx!j jy aj if x y The relative Funk metric is a projective weak metric, it is invariant under projective transformations in the sense that if f W RP n! RP n is a projective transformation, then F ;U x; y/ D F f /;f U / f x/; f y//: Observe also that if U R n is a proper convex domain, then F ;U x; y/ D F x; y/ C r F U x; y/: Lemma 112 In the case U D R n, we have F ;U x; y/ D F x; y/: Proof If U D R n, then its boundary is the hyperplane at infinity H 1 and thus D 1 for any x;y 2 jy!j jx!j Recall that there is no preferred hyperplane in projective space Therefore, the classical Funk geometry is a special case of the relative Funk geometry where the englobing domain U RP n is the complement of a hyperplane Such a set U is sometimes called an affine patch 12 Hilbert geometry Definition 121 The Hilbert metric in a proper convex domain R n is defined as H x; y/ D 1 2 F x; y/ C r F y; x// ; where is considered as a subset of an affine patch U RP n This metric is a projective weak metric Note that for x y we have ; H x; y/ D 1 2 log jy bj jx bj jx aj jy aj where a D a x; y/ and b D a y; x/ The expression inside the logarithm is the cross ratio of the points b, x, y, a, therefore the Hilbert metric is invariant by projective transformations

27 Chapter 2 From Funk to Hilbert geometry 59 y x b a Figure 2 The Hilbert metric Note also that the Hilbert metric coincides with half of the relative Funk metric of the domain with respect to itself: H x; y/ D 1 2 F ;x; y/; In the case where is the Euclidean unit ball, the Hilbert distance coincides with the Klein model (also called the Beltrami Cayley Klein model) of hyperbolic space We refer to Sections in [25] for a nice introduction to Klein s model In the case of a convex polytope defined by the linear inequalities j x/<s j, 1 j k, we have 1 H x; y/ D max 1i;jk 2 log si j y/ s i j x/ s j j x/ : s j j y/ Applying our investigation on Funk geometry immediately gives a number of results on Hilbert geometry In particular, applying Proposition 22 we get: Proposition 122 The Hilbert metric in a convex domain R n satisfies the following properties: (a) H x; y/ 0 and H x; x/ D 0 for all x;y 2 (b) H x; z/ H x; y/ C H y; z/ for all x;y;z 2 (c) H is projective, that is, H x; z/ D H x; y/ C H y; z/ whenever z is a point on the affine segment Œx; y (d) The weak metric H is symmetric, that is, H x; y/ D H y; x/ for any x and y (e) The weak metric H is separating, that is, x y H) H x;y/>0, if and only if the domain does not contain any affine line (f) The weak metric H is unbounded The proof of this proposition easily follows from the definitions and from Proposition 22

28 60 Athanase Papadopoulos and Marc Troyanov Proposition 123 If the convex domain R n does not contain any affine line, then H is a metric in the classical sense Furthermore, it is complete A convex domain which does not contain any affine line is called a sharp convex domain From Theorem 71, we deduce the following necessary and sufficient condition for the equality case in the triangle inequality: Theorem 124 Let x, y and z be three points in a proper convex domain We have F x; y/ C F y; z/ D F x; z/ if and only if both triples of boundary points a x; y/; a y; z/; a x; z/ and a y; x/; a z; y/; a z; x/ are aligned in RP n From Theorem 76, we obtain: Theorem 125 Let W Œ0; 1! be a path in a sharp convex domain of R n Then is a geodesic for the Hilbert metric in if and only if there exist two faces D ;D C Q@ such that for any t 1 < t 2 in Œ0; 1 we have t 2 / t 1 / 2 C t1 /D C / and t 1 / t 2 / 2 C t2 /D / Recall that C p D/ is the cone at p on the face D, see Equation (73) If is smooth, then it is geodesic if and only if Pt/ 2 C t/ D C / and Pt/ 2 C t/ D / for any t 2 Œ0; 1 We then have the following characterization of smooth geodesics in Hilbert geometry which we formulate only for bounded domains for convenience: Corollary 126 Let W Œ0; 1! be a path of class C 1 in a bounded convex domain of R n Then is a geodesic for the Hilbert metric in if and only if there exist two faces D ;D C such that for any t, the tangent line to the curve at t meets the on D C [ D Corollary 127 (compare [10]) Assume that is strictly convex, or more generally that all but possibly one of its proper faces are reduced to points Then the Hilbert geometry in is uniquely geodesic 13 Related questions In this section we briefly discuss some recent developments related to the idea of the Funk distance Yamada recently introduced what he called the Weil Petersson Funk metric on Teichmüller space (see [27], and see [15] in this volume) The definition is analogous to one of the definitions of the Funk metric, using in an essential way the noncompleteness of the Weil Petersson metric on Teichmüller space One can wonder whether there are Funk-like metrics associated to other interesting known symmetric