Weak Finsler Strutures and the Funk Metric

WEAK FINSLER STR UTURES AND THE FUNK METRIC A THANASE P AP ADOPOULOS AND MARC TRO Y ANOV Abstract. W e discuss general notions of m etrics and of Finsler structures which we call we ak metrics and we ak Finsler structur es . Any con vex domain carries a canonical wea k Finsler structure, whic h we call its t autolo gic al w e ak Finsler s tructur e . W e compute distances in the tautological weak Finsler struc- ture of a domain and we sho w that these are giv en by the so-called F unk we ak metric . W e conclude the paper with a discussion of geo desics, of m etric balls and of con vex ity prop erties of the F unk w eak metric. AMS Mathematics Sub ject Classification: 52A, 53C60, 58B20 Keywords: Finsler structure, weak metric, F unk weak metric. Contents 1. Int ro duction 1 2. Preliminarie s o n conv ex geometry 2 3. The notion of weak metric 4 4. W eak length spa ces 4 5. W eak Finsler structures 5 6. The tautologic a l w eak Finsler structure 7 7. The F unk weak metric 10 8. On the geometry o f the F unk weak metric 11 References 17 1. Introduction A we ak metric on a s et is a function defined on pairs o f p oints in that set which is nonnegative, which can take the v alue ∞ , whic h v anishes when the tw o p o ints co - incide and which satisfies the triangle ineq uality . Compared to an or dinary metric, a weak metric can th us degenera te and take infinite v alues. Besides, it need not b e symmetric. This is a very general no tion which turns out to be us eful in v a rious situations. The terminology “weak metric” is due to Rib eiro [20], but the notion can b e at least traced bac k to the work of Hausdor ff (see [1 4]). In the pap e r [18], a nu mber of natura l weak metr ic s ar e discussed. In the pre s ent pap er , we a re mostly int erested in a cla ss o f weak metrics that is related to conv ex geometry and to a general notion of Finsler structures on ma nifolds. A basic construction in c o nv ex geometry is the notion o f Minkowski norm , which asso ciates to an y conv ex set containing the or igin in a vector space V a tra nslation- inv a riant homogeno us w eak metric on V . Finsler geometry is an extension of this Date : April 3, 2008. 1 2 A THANASE P AP ADOPO ULOS AND MARC TR OY ANOV construction to an a rbitrary manifold. W e define a we ak Finsler strucure on a differentiable manifold to b e a field o f conv ex sets on that manifold. More precisely , a w eak Finsler strucure is a subse t of the tangent s pace of the manifold whose int ersec tio n with ea ch fib er is an con vex set con taining the origin. The Mink owski norm in each tangent space of a manifold e ndow ed with a weak Finsler structure gives rise to a function defined on the tota l space of the tang e nt bundle. W e call this function the L agr angian of the w eak Finsler struc tur e. Integrating this Lagrangia n on piecewise smo oth cur ves in th e manifold defines a length str ucture and thus a notion of dista nce on the manifold. This distance is genera lly a weak metric. A case of s pec ial in terest is when the manifold is a con vex domain Ω in R n and when the weak Finsler structure is obtained by replica ting at each p oint of Ω the doma in Ω itself. W e ca ll this the tautolo gic al weak Finsle r s tructure, and we study some of its basic prop erties in the present pap er. More pre c isely , we first give a formula for the distance b etw ee n tw o p oints. It turns out that this dis ta nce coincides with the metric intro duced by P . F unk in [13]. W e then study the geometry o f balls and the geo desics in the F unk w eak metric. Mo dern r eferences on Finsler geometry include [9], [2], [3], and [1]. One of Herb ert Busemann’s ma jor ideas, expres sed in [5], [6], [7] and [8] is that Finsler geometry should b e develop ed without lo cal co ordinates and witho ut the use of differ ent ial calculus. This paper brings s ome results in that direction. 2. Preliminaries on convex geometr y In this section, we re c a ll a few notions in co nv e x geometr y that will b e use d in the sequel. Given a conv ex subset Ω of R n , we shall denote Ω its closure, o Ω its interior, a nd ∂ Ω = Ω \ o Ω its b oundar y . Let Ω ⊂ R n be a (not neces sarily op en) con vex set and let x b e a point in Ω. Definition 2.1 . The r adial function of Ω with r esp e ct to x is the function r Ω ,x : R n → R + ∪ {∞} defined b y r Ω ,x ( ξ ) = sup { t ∈ R | ( x + tξ ) ∈ Ω } . Definition 2.2. The Minkowski function of Ω with r esp e ct to x is the function p Ω ,x : R n → R + ∪ {∞} defined b y p Ω ,x ( ξ ) = 1 r Ω ,x ( ξ ) . Classically , the Minko wski function is a sso ciated to an op en co nv ex subset Ω of R n containing the or igin 0, and ta king x = 0. T his function is so metimes calle d the Minkowski we ak norm of the con vex (see e.g. [10], [1 7], [21] and [2 2]). W e a lso recall t hat for an y conv ex set Ω in R n , there exists a w ell-defined smalle s t affine subspace L of R n containing Ω, and that the intersection of Ω with L has nonempty in terior in L . W e denote by RelIn t(Ω) this interior, ca lled the r elative interior of the c onvex set Ω. The following prop o sition collects a few basic prop erties of the Minkowski function. In particular, Proper ty (8) t ells us that w e can reconstruct the relative interior of Ω from the Mink owski function of Ω at any p oint. The proofs are eas y . WEAK FINSLER STRUTURES AND THE FUNK METRIC 3 Prop ositi o n 2.3. L et Ω b e a c onvex su bset of R n . F or every x in Ω and for every ξ and η in R n , we have (1) p Ω ,x ( ξ ) = inf { t ≥ 0 | ξ ∈ t (Ω − x ) } . (Her e, Ω − x denotes the Minkowski sum of Ω and − x .) (2) If the r ay { x + tξ | t ≥ 0 } is c ontaine d in Ω , then p Ω ,x ( ξ ) = 0 . (3) p Ω ,x ( λξ ) = λp Ω ,x ( ξ ) for λ ≥ 0 . (4) p Ω ,x ( ξ + η ) ≤ p Ω ,x ( ξ ) + p Ω ,x ( η ) . (5) The Minkowski function p Ω ,x is c onvex. (6) If x is in o Ω , then p Ω ,x is c ontinuous. (7) If Ω i s close d, t hen Ω = { y ∈ R n | y = x + ξ, p Ω ,x ( ξ ) ≤ 1 } . (8) RelInt (Ω) = { y = x + ξ | p Ω ,x ( ξ ) < 1 } . (9) If Ω 1 = RelInt(Ω) , then p Ω 1 ,x = p Ω ,x . In some cas es, w e can giv e ex plic it formulas for the Minko wski function p Ω ,x . F or instance, the Mink owski function of the closed ball B = B (0 , R ) in R n of r a dius R and center 0 with r esp ect to an y p oint x in B is given b y p B ,x ( ξ ) = p h ξ , x i 2 + ( R 2 − | x | 2 ) | ξ | 2 + h ξ , x i ( R 2 − | x | 2 ) . The Minko wski function of a half-space H = { x ∈ R n   h ν, x i ≤ s } , wher e ν is a vector in R n (whic h is orthogo nal to the h yp erplane b ounding H ) and where s is a real num b er , with respect to a p oint x in H , is giv en b y p H,x ( ξ ) = max  h ν, ξ i s − h ν, x i , 0  . W e shall use this f ormula later o n in this paper . W e also recall the follo wing: Definition 2.4 (Support h yp erplane) . Let Ω b e a nonempty subset of R n . An a ffine hyperplane A in R n is ca lled a supp ort hyp erplane for Ω if the r e lative interior of Ω is contained in one of the tw o closed half-spa ces b ounded b y A and if Ω ∩ A 6 = ∅ . If A is a supp or t h y pe r plane f or Ω and if x is a point in Ω ∩ A , then A is ca lled a supp ort hyp erplane for Ω at x . When Ω ⊂ R 2 , then A is called a s u pp ort line . Suppo se now that Ω a convex subs e t of R n . It is known that any p oint on the bo undary of Ω is con tained in at least one of its suppor t hyperplanes (see e .g . [10] p. 20). The intersection of Ω with any of its support hyper planes is a conv ex set which is nonempty if Ω is clo sed. This intersection is not a lwa ys reduced to a p oint. W e reca ll the notion of a strictly conv ex subset in R n , and b efore that we note the following clas sical prop ositio n: Prop ositi o n 2.5. L et Ω b e an op en c onvex subset of R n . The n, the fol lowing ar e e quivalent: (1) ∂ Ω do es not c ontain any nonempty op en affine se gment; (2) e ach supp ort hyp erplane of Ω i nterse cts ∂ Ω in exactly one p oint; (3) supp ort hyp erplanes at distinct p oints of ∂ Ω ar e distinct; (4) any line ar function on R n has exactly one maximu m on ∂ Ω . Definition 2.6 (Strictly conv ex subset) . Le t Ω b e an op en co nv ex subset of R n . Then, Ω is said to b e strictly c onvex if one (or, equiv alently , all) t he properties of Prop ositio n 2.5 are sa tisfied. 4 A THANASE P AP ADOPO ULOS AND MARC TR OY ANOV 3. The notion of weak metric Definition 3. 1. A we ak metric on a set X is a function δ : X × X → R + ∪ { ∞} satisfying (1) δ ( x, x ) = 0 for all x in X ; (2) δ ( x, z ) ≤ δ ( x, y ) + δ ( y , z ) for all x , y and z in X . W e say that such a weak metr ic δ is symmetric if δ ( x, y ) = δ ( y , x ) for all x and y in X ; that it is fi nite if δ ( x, y ) < ∞ for every x and y in X ; that δ is st r ongly sep ar ating if we have the eq uiv alence min( δ ( x, y ) , δ ( y , x )) = 0 ⇐ ⇒ x = y ; and that δ is we akly sep ar ating if we hav e the equiv alence max( δ ( x, y ) , δ ( y , x )) = 0 ⇐ ⇒ x = y . W e r ecall that the notion o f weak metric already appea rs in the work of Hausdorff (cf. [14], in w hich Hausdo rff defines asy mmetric distances on v a rious sets of subsets of a metric space). Definition 3.2 (Geo desic) . Let ( X, δ ) b e a weak metric space and let I ⊂ R b e an interv al. W e say tha t a m ap γ : I → X is ge o desic if for every t 1 , t 2 and t 3 in I satisfying t 1 ≤ t 2 ≤ t 3 we have δ ( γ ( t 1 ) , γ ( t 2 )) + δ ( γ ( t 2 ) , γ ( t 3 )) = δ ( γ ( t 1 ) , γ ( t 3 )) . W ea k metric s w ere extensively studied by Buse ma nn, cf. [5], [6], [7] & [8]. A basic example of a w eak metric defined on a co nv ex set in R n is the follo wing: Example 3.3. Let Ω ⊂ R n be a c onv ex set suc h that 0 ∈ Ω and let p ( ξ ) = p Ω , 0 ( ξ ) = inf { t > 0 | ξ ∈ t Ω } b e the Minko wski weak no r m centered at 0 o f Ω. Then, the function δ : R n × R n → R + ∪ {∞} defined b y δ ( x, y ) = p ( y − x ) is a weak metric o n R n . F or this w eak metric, w e hav e the following equiv a lences: (1) δ is finite ⇐ ⇒ 0 ∈ o Ω ; (2) δ is symmetr ic ⇐ ⇒ Ω = − Ω; (3) δ is stro ngly separa ting ⇐ ⇒ Ω is bounded; (4) δ is weakly separating ⇐ ⇒ Ω does not co nt ain a ny E uc lide a n line. The weak metric on R n defined in Example 3.3 is called t he Minkowski we ak met- ric asso ciate d to Ω. The a sso ciated weak metric s pa ce ( R n , δ ) is called a we ak Minkowski sp ac e . 4. Weak length sp aces Let X be a set a nd let Γ b e a g roup oid o f paths in X . Conca tenation of pa ths is de- noted b y the sym b ol ∗ . The in verse γ − 1 of a path γ : [ a, b ] → X is the path obtained by pre-co mp o s ing γ with the unique affine sense-reversing self-homeo morphism of [ a, b ]. Definition 4.1. A we ak length stru ctur e on ( X , Γ) is a function ℓ : Γ → R + ∪ {∞} which sa tis fies the following pro pe r ties: WEAK FINSLER STRUTURES AND THE FUNK METRIC 5 (1) Invarianc e un der r ep ar ametrization : if [ a, b ] a nd [ c, d ] ar e interv als of R , if γ : [ a, b ] → X is a path in X that b elongs to Γ and if f : [ c, d ] → [ a, b ] is an increasing homeo morphism such that γ ◦ f ∈ Γ, then ℓ ( γ ) = ℓ ( γ ◦ f ). (2) A dditivity : for ev ery γ 1 and γ 2 in Γ, we hav e ℓ ( γ 1 ∗ γ 2 ) = ℓ ( γ 1 ) + ℓ ( γ 2 ). A weak length structure Γ is sa id to be r eversible if for e very γ in Γ, γ − 1 is also in Γ and w e ha ve ℓ ( γ − 1 ) = ℓ ( γ ). A weak length structure Γ is s aid to be sep ar ating if w e hav e th e equiv alence: ℓ ( γ ) = 0 ⇐ ⇒ γ is a unit in Γ (i.e. γ is a consta n t path). Let ( X , Γ , ℓ ) b e a set equipp ed with a g r oup oid of paths and with a weak length structure. W e s et (4.1) δ ℓ ( x, y ) = inf γ ∈ Γ x,y ℓ ( γ ) , where Γ x,y = { γ ∈ Γ | γ joins x to y } . It is ea sy to s ee that t he fun ction δ ℓ is a w eak metric o n X . Definition 4. 2. Let ( X, Γ , ℓ ) be a set equipp ed with a group oid of paths and with a weak length structure. The weak metric δ ℓ defined in (4.1) is called the we ak metric asso ciate d to t he we ak length st ructur e ℓ . A we ak length met ric sp ac e is a weak metric space X obtained from such a triple ( X, Γ , ℓ ) by equipping X with the asso ciated weak metric δ ℓ . 5. Weak Finsler structures W e in tro duce a genera l notion of Finsler structure, which we call we ak Finsler structur e , and which can b e c o nsidered as an infinitesimal notion of weak length structure. Definition 5.1. Let M b e a C 1 manifold a nd let T M b e its tang ent bun dle. A we ak Finsler structu r e on M is a subset e Ω ⊂ T M such that for eac h x in M , the subset Ω x = e Ω ∩ T x M of the t angent space T x M of M at x is c o nv ex and con tains the or igin. W e pr ovide the set of all weak Finsler structures on M with the or der relation  defined as follo ws: f Ω 1  f Ω 2 ⇔ f Ω 1 ⊃ f Ω 2 . Examples 5. 2. In the following examples, M is a C 1 manifold. (1) e Ω = T M is a weak Finsler structure, which we ca ll the minimal weak Finsler structure. (2) e Ω = M ⊂ T M , e mbedded a s the zero section, is a weak Finsler structure which w e call th e maximal weak Finsler structure. (3) If e Ω and f Ω ′ are t wo Finlser structures on M , then e Ω ∩ f Ω ′ ⊂ T M is also a Finsler structure. (4) If e Ω and f Ω ′ are tw o Finlser structures on M , then, taking the unio n of the Minko wski sums Ω x + Ω ′ x of the conv ex sets in each tangent space T x M , we obtain the Minkowski su m Finsler structur e e Ω + f Ω ′ ⊂ T M . (5) If ω is a differen tial 1 -form on M , then e Ω ω = { ( x, ξ ) ∈ T M | ω x ( ξ ) ≤ 1 } 6 A THANASE P AP ADOPO ULOS AND MARC TR OY ANOV and e Ω | ω | = { ( x, ξ ) ∈ T M | | ω x | ( ξ ) ≤ 1 } are weak Finsler structures on M . (6) If ω and ω ′ are t wo 1 -froms o n M , then max( ω , ω ′ ) defines a w eak F inlser structure on M . (7) If e Ω is a weak Fin lser structure o n M and if N ⊂ M is a C 1 submanifold, then e Ω N = e Ω ∩ T N is a weak Finls e r structure on N , called the weak Finsler structure induc e d by the embedding N ⊂ M . (8) If e Ω is a weak Finlser structures on M , if N is a C 1 manifold and if f : N → M is a C 1 map, then ( T f ) − 1 ( e Ω) ⊂ T N is a Finsler structure on N . W e denote it by f ∗ ( e Ω) and call it the p ul l b ack of e Ω by the map f . Definition 5.3 (La grangia n) . The La gr angian of a weak Finlser structure e Ω on a C 1 manifold M is the function on the tangent bundle T M who se restriction to each tangent space T x is the Mink owski function of Ω x . It is th us defined b y p ( x, ξ ) = p e Ω ( x, ξ ) = inf { t | t − 1 ξ ∈ Ω x } . The qua n tity p ( x, ξ ) is a lso called the Fi nsler norm of the vector ( x, ξ ) relative to the given w eak Finls e r structure. Example 5.4. Let g b e a Riemannian metric on M , let ω is a differential 1-form and let µ b e a smo oth function on M satisfying | µω x | < 1 at every po int x in M . Then, p = √ g + µω is the Lagra ngian of a Finsler structure on M . Such a Finlser structure is usually called a R anders metric on M , and it has applicatio ns on physics (cf. e.g. [3] § 11.3, and see also [4] for the relation of this metric with the Zer melo navigation pr oblem.) Lemma 5.5. L et e Ω b e a we ak Finlser st ructur e on M . Assume that M (c onsider e d as a subset of T M - the zer o se ction) is c ontaine d in t he interior of e Ω ⊂ T M . Then the asso ciate d L agr angian p : T M → R is upp er semi-c ontinuous. Pr o of. The hypo thesis implies tha t for every x in M , the interior of each conv ex set Ω x = e Ω ∩ T x M ⊂ T x M is nonempty . Therefore, the usual in terior and the relative interior of Ω x coincide. Prop erty (9) of P rop osition 2.3 implies then that the Lag rangian of e Ω co incide s with the Lag r angian of its in ter ior Int  e Ω  . One may therefore a ssume without loss of generality that e Ω ⊂ T M is an op en set, and in pa rticular e Ω = { ( x, ξ ) ∈ T M   p ( x, ξ ) < 1 } (see Pro po sition 2.3 (8)). Now for any t ∈ R , the sublevel s et { p ( x, ξ ) < t } is either empt y (when t ≤ 0 ) o r it is homothetic to the o pen s et e Ω ⊂ T M (when t > 0). In any case, it is an op en s ubset o f T M , a nd p : T M → R is therefore upp e r semi-contin uous.  Prop ositi o n 5.6. L et e Ω b e a Finsler st ructur e on a C 1 manifold M and let p e Ω : T M → R b e the asso ciate d L agr angian. Then, (1) for every x in M , t he function ξ 7→ p ( x, · ) is a we ak norm on T x M ; WEAK FINSLER STRUTURES AND THE FUNK METRIC 7 (2) if f Ω ′ ⊂ T M is another Finsler structur e o n M , with asso ciate d L agr angian p f Ω ′ , then we have the e quivalenc e e Ω  f Ω ′ ⇐ ⇒ p e Ω ≤ p f Ω ′ , (3) p e Ω : T M → R is B or el-me asur able. Pr o of. The fir st tw o assertions ar e ea sy to chec k and we only prove the last one. If M is contained in the in ter ior of e Ω ⊂ T M , then, by Lemma 5.5, the Lagra ng ian p is upp er semi-contin uous and therefo r e Bo rel measura ble. In the gener al case, M is contained in e Ω but not necessar ily in its int erio r . W e consider a decr e a sing sequence T M  e Ω 1  e Ω 2  · · ·  e Ω of weak Finsler structures such that M is contained in the interior of e Ω j ⊂ T M for every j ∈ N and e Ω = ∞ \ j =1 e Ω j W e then hav e p e Ω 1 ≤ p e Ω 2 ≤ · · · ≤ p e Ω and p e Ω = sup j p e Ω j = lim j →∞ p e Ω j Therefore p e Ω is the limit of a seq uence of Bor el measura ble functions a nd is thus Borel measura ble.  W e shall say that the Finlser structure e Ω is smo oth if p is smo oth. Definition 5. 7 (The weak length structure asso ciated to a w eak Finsler structure) . Let M be a C 1 manifold equipp ed with a weak Finlser structure e Ω with Lagr a ngian p . There is an asso cia ted w e ak length structure on M , defined by taking Γ to b e the gr oup oid of piecewis e C 1 paths, a nd defining, f or eac h γ : [ a, b ] → M in Γ, (5.1) ℓ ( γ ) = Z b a p ( γ ( t ) , ˙ γ ( t )) dt. Remark 5.8. In Equation (5.1), γ and ˙ γ ar e contin uous, and since p is Bor el- measurable, the map t 7→ p ( γ ( t ) , ˙ γ ( t )) is nonnega tive and measur able. Therefore, the Leb esg ue in tegral is well defined. 6. The t autological weak Finsler structure In this section, Ω is a n op en con vex s ubs e t of R n . W e shall use the natural iden ti- fication T Ω ≃ Ω × R n . Definition 6.1 (The tautolog ical weak Finsler structure ) . The taut olo gic al we ak Finsler structur e on Ω is the w ea k Finsler structure e Ω ⊂ T Ω defined b y e Ω = { ( x, ξ ) ∈ Ω × R n | x ∈ Ω and x + ξ ∈ Ω } . This structure is called “tautologica l” because the fibre over eac h p oint x of Ω is the set Ω itself (with the origin at x ). The pro of o f next propos ition follows ea sily from t he definitions. 8 A THANASE P AP ADOPO ULOS AND MARC TR OY ANOV Prop ositi o n 6.2 . L et Ω b e an op en c onvex subset of R n e quipp e d with its tauto- lo gic al we ak Finsler stru ctur e e Ω . Then, for every x in Ω , the Finsler norm of any tangent ve ctor ξ at x is given by p Ω ,x ( ξ ) , wher e p Ω ,x is the Minkowski funct ion of Ω wi th r esp e ct to x . Given an op en conv ex subset Ω of R n , we denote by d Ω the weak length metric asso ciated to the tautolo gical weak Finsler s tr ucture o n Ω. This weak metric is th us defined b y (6.1) d Ω ( x, y ) = inf γ ∈ Γ x,y Z γ p ( γ ( t ) , ˙ γ ( t )) dt. where Γ x,y is the s et of piec e wise C 1 paths joining x to y . Lemma 6.3 . L et Ω and Ω ′ b e two c onvex op en subsets of R n satisfying Ω ⊂ Ω ′ , then d Ω ′ ≤ d Ω . In the re s t of this pap er, we shall use the follo wing notations: F or x and y in R n , we denote by | x − y | their E uclide a n distance. Giv en tw o distinct points x and y in Ω, R ( x, y ) denote the Euclidean ray star ting at x and pa ssing through y . In the case where R ( x, y ) 6⊂ Ω w e set a + = a + ( x, y ) = R ( x, y ) ∩ ∂ Ω. Theorem 6 .1. L et Ω b e a n op en c onvex s u bset of R n e quipp e d with its tautolo gic al we ak Finsler s t ructur e. Then, for every x and y in Ω , the Euclide an se gment c onne cting x and y is of minimal length, and the asso ciate d we ak metric on Ω is given by d Ω ( x, y ) =    log | x − a + | | y − a + | if x 6 = y and R ( x, y ) 6⊂ Ω 0 otherwise . Pr o of. As b efore, w e let d Ω denote the weak metric defined b y the ta uto logical weak Finsler structure on Ω. W e also denote by ℓ ( γ ) the length of a path γ for the tautological weak Finsler weak length s tructure. The pro of o f the theorem is done in four steps. Step 1.— Suppo se that R ( x, y ) ⊂ Ω. Consider the linear pa th γ : [0 , | x − y | ] → Ω defined by (6.2) γ ( t ) = x + t y − x | y − x | . The deriv ative of the path γ is the constan t vector ˙ γ ( t ) = y − x | y − x | . Therefore, p e Ω ( γ ( t ) , ˙ γ ( t )) = 1 | y − x | p e Ω ( γ ( t ) , y − x ), which is equal to 0 since R ( x, y ) ⊂ Ω. Now the path γ has length zer o and satisfies γ (0) = x and γ ( | y − x | ) = y . Ther efore d Ω ( x, y ) = 0. In the r e st of this pro of, we suppos e that R ( x, y ) 6⊂ Ω. WEAK FINSLER STRUTURES AND THE FUNK METRIC 9 Step 2.— W e show that for e very distinct p oints x and y in Ω and for every Eu- clidean segment γ joining x to y , w e hav e (6.3) d Ω ( x, y ) ≤ ℓ ( γ ) = lo g | x − a + | | y − a + | . Using the radial function r Ω ,x int ro duced in § 2, we ca n write a + = a ( x, y − x ) = x + r Ω ,x ( y − x ) · ( y − x ) . T o compute the Finsler leng th of the Euclidea n seg men t [ x, y ], we parametrize it as the path γ defined in (6.2). F o r 0 ≤ t ≤ | x − y | , le t r ( t ) = | x − γ ( t ) | . Then, r ( t ) = r Ω ,x ( γ ( t ) , ˙ γ ( t )), a nd it is easy to see tha t r ( t ) = | x − a + | − t. Then, we hav e r ′ ( t ) = − 1 and therefore ℓ ( γ ) = Z | y − x | 0 dt r ( t ) = − Z | y − x | 0 r ′ ( t ) dt r ( t ) = − log  r ( t )     t = | y − x | t =0 = log | x − a + | | y − a + | . This gives the desir ed inequality (6.3). Step 3.— W e complete the pro of of the theor e m in the particular ca se wher e Ω is a half-space. By t he in v arianc e of t he tautological Finsler structure under the group of affine t ra ns formations, it suffices to co ns ider the case wher e Ω is the half-space H ⊂ R n defined by the equatio n H = { x ∈ R n   h ν, x i ≤ s } , for some vector ν in R n (whic h is orthogonal to the h yp erplane bo unding H ) and for some s in R . Rec all that the Minkowski fun ction as so ciated to H is given b y the formula p H ( x, ξ ) = max  h ν, ξ i s − h ν , x i , 0  . Consider now an arbitra ry piecewise C 1 path α : [0 , 1] → H suc h that x = α (0) and y = α (1). Then, ℓ ( α ) = Z 1 0 max  h ν, ˙ α ( t ) i s − h ν , α ( t ) i , 0  dt ≥ Z 1 0 h ν, ˙ α ( t ) i s − h ν, α ( t ) i dt. W e ha ve h ν, ˙ α ( t ) i s − h ν , α ( t ) i = − d dt  log  s − h ν, α ( t ) i  . Therefore, ℓ ( α ) ≥ − log  s − h ν , α (1) i  + log  s − h ν, α (0) i  = lo g s − h ν, x i s − h ν, y i . Now we note that s − h ν, x i = s − h ν, x − a + i − h ν, a + i = h x − a + , − ν i = h ν, a + − x i . Likewise, s − h ν , y i = h ν, a − y i . Thu s, we obtain ℓ ( α ) ≥ log h ν, a + − x i h ν, a + − y i . 10 A THANASE P AP ADOPO ULOS AND MARC TR OY ANOV Now using the fac t that the thre e points x, y , a + are aligned in that order and that ν is not parallel to the v ector x − y , we easily see that h ν, a − x i h ν, a − y i = | x − a + | | y − a + | , which gives ℓ ( α ) ≥ log | x − a + | | y − a + | . Since α was ar bitr ary , w e have d H ( x, y ) ≥ log | x − a + | | y − a + | . Combining this ineq uality and the inequality (6 .3 ), we obta in, in the ca s e where Ω = H is a ha lf-s pace, d H ( x, y ) = log | x − a + | | y − a + | . In particular an y Euclidean seg ment is leng th minimizing. Step 4. — No w we prov e the prop o s ition for a general op e n convex set Ω. Let x and y be t wo elements in Ω a nd consider the Euclidean ray R ( x, y ). By hypothesis, w e have L 6⊂ Ω, and as before, we set a + = R ( x, y ) ∩ ∂ Ω. W e let A denote a suppo rt hyper plane to Ω throug h a + , and we let H b e the open half-spa ce containing Ω and who s e bo unda ry is equal t o A . Using Lemma 6 .3 a nd Step 3, we hav e d Ω ( x, y ) ≥ d H ( x, y ) = log | x − a + | | y − a + | . Combining this with the inequality (6.3) w e obtain d Ω ( x, y ) = log | x − a + | | y − a + | . The argument also prov es that any Euclidea n segment γ is le ng th minimizing. This completes the proof o f Theorem 6 .1.  7. The Funk weak metric In this a nd the following section, we give a quick overview of the F unk weak metric, of its g eo desics, of its balls and o f its topolog y . The F unk w eak metric is a nice example of a weak metric, and a geometric study of this w eak metric is so mething whic h seems missing in the litera ture. W e study this weak metric in mo re detail in [19]. In this section, Ω is a nonempty open conv ex subset o f R n . W e use the notations a + , R ( x, y ), etc. established in the pr e ceding section. Definition 7.1 (The F unk w eak metric) . The F u nk we ak metric of Ω, denoted by F Ω , is d efined, for x and y in Ω, b y th e form ula F Ω ( x, y ) =    log | x − a + | | y − a + | if x 6 = y and R ( x, y ) 6⊂ Ω 0 otherwise . WEAK FINSLER STRUTURES AND THE FUNK METRIC 11 Observe that Theorem 6 .1 says that the F unk w eak met ric is the weak metric asso ciated to the tautological Finsler structure in Ω. In par ticular the tria ngle inequality is verified. Another pro of o f the triangle ineq uality is given in [23] p. 85. This proof is not trivia l and uses a rguments similar to those o f the classical proof of the tr ia ngle inequality for the Hilbert met ric, as given by D. Hilb ert in [16]. If Ω = R n , then F ≡ 0. W e sha ll henceforth assume that Ω 6 = R n whenever we shall dea l with the F unk weak metric of an nonempty open convex subset Ω of R n . The F unk w eak metric is always unbounded. Indeed, if x is an y point in Ω and if x n is any sequence of p oints in that s pa ce conv erging to a p oint on ∂ Ω (conv erg e nce here is with r esp ect to the Euclidean metric), then F Ω ( x, x n ) → ∞ . Notice that on the other hand F Ω ( x n , x ) is bounded. Example 7.2 (The upper half- pla ne) . Let Ω = H ⊂ R 2 be the upper half-plane, that is, H = { ( x 1 , x 2 ) ∈ R 2 | x 2 > 0 } . Then, for x = ( x 1 , x 2 ) and y = ( y 1 , y 2 ) in H , we hav e F H ( x, y ) = max  log x 2 y 2 , 0  . The following three pr op ositions are easy conse q uences of the definitions and th ey will b e use d below. W e take Ω to b e again a no nempt y op en subset of R n . Prop ositi o n 7.3. Le t Ω ′ ⊂ Ω b e the int erse ction of Ω with an affine subsp ac e of R n , and su pp ose that Ω ′ 6 = ∅ . Then, F Ω ′ is the we ak metric ind uc e d by F Ω on Ω ′ . Prop ositi o n 7. 4. In t he c ase wher e Ω is b ounde d, t he F unk we ak metric F Ω is str ongly sep ar ating, and we have the f ol lowing e quivalenc es: (7.1) F Ω ( x, x n ) → 0 ⇐ ⇒ F Ω ( x n , x ) → 0 ⇐ ⇒ | x − x n | → 0 . Prop ositi o n 7.5. L et Ω 1 and Ω 2 b e two op en c onvex subsets of R n . Then, F Ω 1 ∩ Ω 2 = max { F Ω 1 , F Ω 1 } . 8. On the geometr y o f the Funk weak metric In this section, w e study the geodes ics, and then, the geometric balls of the F unk weak metric. Prop ositi o n 8.1. L et x , y a nd z b e thr e e p oints in Ω lying in that or der on a Euclide an line. Then, we have F ( x, y ) + F ( y , z ) = F ( x, z ) . This results follo ws from Theorem 6.1, but it is also quite simple to prov e it directly . Pr o of. W e can ass ume that the three p oints are distinct, otherwise the pro of is trivial. W e hav e R ( x, y ) ⊂ Ω ⇐ ⇒ R ( x, z ) ⊂ Ω ⇐ ⇒ R ( y , z ) ⊂ Ω, a nd this holds if a nd o nly if the three quantities F ( x, y ), F ( y , z ) and F ( x, z ) are equa l to 0. Thus, the conclusion also holds trivially in this c ase. Therefo r e, we can assume that R ( x, y ) 6⊂ Ω. In this cas e , we ha ve a + ( x, y ) = a + ( x, z ) = a + ( y , z ). Denoting this common point by a + , we hav e | x − a + | | y − a + | | y − a + | | z − a + | = | x − a + | | z − a + | , 12 A THANASE P AP ADOPO ULOS AND MARC TR OY ANOV which implies log | x − a + | | y − a + | + log | y − a + | | z − a + | = log | x − a + | | z − a + | , which completes the pro of.  Corollary 8.2. The Euclide an se gments in Ω ar e ge o desic se gments for the F unk we ak metric on Ω . Since the op en set Ω is conv ex, Co rollar y 8.2 implies that (Ω , F Ω ) is a geo desic weak metric space (any tw o p oints can b e joined by a geo de s ic seg ment) . It a lso says that (Ω , F Ω ) is a Desarguesian space in the sense of H. Busemann (see [7 ]). Notice that in general, the E uclidean segments are not the only geo desic segments for a F unk weak metric. In f act, the follo wing propositio n implies that th ere exist other types of geo desic segments in Ω, provided there exists a Euclidea n segment of nonempty interior c o ntained in the b o undary o f Ω. Prop ositi o n 8.3. L et Ω b e an op en c onvex subset of R n such that ∂ Ω c ontains a Euclide an se gment [ p, q ] and let x and z b e two p oints in Ω such that R ( x, z ) ∩ [ p, q ] 6 = ∅ . L et Ω ′ b e the interse ction of Ω with the affine su bsp ac e of R n sp anne d by { x } ∪ [ p, q ] . Then, for any p oint y in Ω ′ satisfying R ( x, y ) ∩ [ p, q ] 6 = ∅ and R ( y , z ) ∩ [ p, q ] 6 = ∅ , we ha ve F ( x, y ) + F ( y , z ) = F ( x, z ) . P S f r a g r e p la c e m e n t s p q x y z x ′ y ′ z ′ Figure 1. Pr o of. It suffices to w ork in the space Ω ′ . Let x ′ , y ′ and z ′ denote the feet of the per p endiculars fro m x and z respectively on the Euc lide a n line joining the po ints p and q (se e Figure 1). Le t b = R ( x, z ) ∩ [ p , q ]. Since the triangles bxx ′ and b z z ′ are similar, we hav e F ( x, z ) = lo g | x − b | | z − b | = lo g | x − x ′ | | z − z ′ | . WEAK FINSLER STRUTURES AND THE FUNK METRIC 13 Similar formulas hold for F ( x, y ) a nd F ( y , z ). Ther efore, F ( x, z ) = log | x − x ′ | | z − z ′ | = log  | x − x ′ | | y − y ′ | | y − y ′ | | z − z ′ |  = log  | x − x ′ | | y − y ′ |  + log  | y − y ′ | | z − z ′ |  = F ( x, y ) + F ( y , z ) .  Remark 8.4. By taking limits o f polyg onal paths, we can easily construct, from Prop ositio n 8.3, smo oth paths whic h are not E uclidean paths a nd which are geo desic for the F unk weak metric. Prop ositi o n 8.5. L et Ω b e an op en c onvex subset of R n . L et x and z b e t wo distinct p oints in Ω su ch that R ( x, z ) ∩ ∂ Ω 6 = ∅ and such that at the p oint b = R ( x, z ) ∩ ∂ Ω , ther e is a supp ort hyp erplane who se i nterse ction with ∂ Ω is r e duc e d to b . L et y b e a p oint in Ω such t hat the thr e e p oints x, y , z in Ω do not lie on the same affine line. Then, F ( x, z ) < F ( x, y ) + F ( y , z ) . Pr o of. T o pro ve the propo sition, w e w ork in the a ffine plane spanned by x , y a nd z and therefo r e we can assume witho ut loss of gener ality that n = 2 . W e a ssume that the in ter section p oints of R ( x, y ) and R ( y , z ) with ∂ Ω are not empt y , and we let a and c b e resp ectively these p oints. F rom the h yp othesis, there is a supp or t line of Ω (whic h we ca ll D ) at b whose in tersectio n with ∂ Ω is reduced to the p o int b . F o r the pro of, we distinguish thr ee cases. P S f r a g r e p la c e m e n t s D a ′ a x y z c b c ′ Figure 2. 14 A THANASE P AP ADOPO ULOS AND MARC TR OY ANOV Case 1.— The tw o rays R ( x, y ) a nd R ( y , z ) intersect the line D (see Fig ure 2). Let a ′ and c ′ be resp ectively these intersection p o int s. Note that the t hree p oints a ′ , b and c ′ are in that o r der on D . By rea s oning with pro jections on the line D and arg uing as we did in t he proof o f Pr op osition 8.3, we hav e | x − b | | z − b | = | x − a ′ | | y − a ′ | | y − c ′ | | z − c ′ | . Since we hav e | x − a ′ | | y − a ′ | < | x − a | | y − a | and | y − c ′ | | z − c ′ | < | y − c | | z − c | , we obta in | x − b | | z − b | < | x − a | | y − a | | y − c | | z − c | which gives, by taking loga rithms, F ( x, z ) < F ( x, y ) + F ( y , z ). Case 2.— The r ay R ( x, y ) intersects D and the ray R ( y , z ) do es not intersect D (Figure 3). W e let as b efore a ′ denote the p o int R ( x, y ) ∩ D . Let D ′ be the Euclidean line passing thro ug h z and parallel to D . The hypotheses in the case considere d imply that the line D ′ int ersec ts the seg ment [ x, y ]. Let y ′ be this in tersectio n p oint. The p oint y ′ is co nt ained in Ω. W e ha ve, as in Case 1, F ( x, z ) = lo g | x − b | | z − b | and F ( x, y ) = lo g | x − a | | y − a | > log | x − a ′ | | y − a ′ | . Now we hav e | x − b | | z − b | = | x − a ′ | | y ′ − a ′ | < | x − a ′ | | y − a ′ | , that is, F ( x, z ) < F ( x, y ), whic h implies t he desired result. Case 3.— The ray R ( x, y ) does not in tersect t he line D . This case can b e treated as C a se 2, and we have in this c ase F ( x, y ) < F ( y , z ), which implies the desired result.  The following is a dir e ct cons e quence of Pr op osition 8.5. Corollary 8.6. Le t Ω b e an op en b ounde d str ictly c onvex subset of R n and let x , y and z b e thr e e p oints in Ω that ar e not c ontaine d in an affine se gment. Then, F ( x, z ) < F ( x, y ) + F ( y , z ) . Corollary 8.7. L et Ω b e an op en b ounde d strictly c onvex subset of R n . Then, the affine se gments in Ω ar e the only ge o desic se gments for t he F unk we ak metric of Ω . Pr o of. This fo llows from the previous Corolla ry and Co rollar y 8 .2, which says that the affine s egments a re geo desic segmen ts for the F unk weak metric.  WEAK FINSLER STRUTURES AND THE FUNK METRIC 15 P S f r a g r e p la c e m e n t s D D ′ a ′ x y z b y ′ Figure 3. W e re c all that a subset Y in a (weak) metric spa ce X is s aid to b e ge o desic al ly c onvex if for any tw o p oints x a nd y in Y , any geo desic se gment in X joining x a nd y is contained in Y . Corollary 8. 8. L et Ω b e an op en b oun de d strictly c onvex subset of R n and let Ω ′ b e a subset of Ω . Then, Ω ′ is c onvex with r esp e ct to the affine structu r e of R n if and only if Ω ′ is a ge o desic al ly c onvex subset of Ω with r esp e ct to t he F unk metric F Ω . Remark 8.9. Note the formal analo gy b etw een Coro llary 8.7 and the following well known result on the geo desic se g ments of a Minkowski metric on R n : if the unit ball of a Minko wski metr ic is stric tly co nvex, then t he only geodesic segments of this metr ic a r e the affine segmen ts. W e now consider spher es a nd balls in a F unk weak metr ic space (Ω , F ). As this weak metric is non-symmetric , we hav e to distinguish b etw een right and left spher es, and we use the following nota tions. F o r an y p oint x in Ω and any nonnega tive real nu mber δ , we set ◦ B ( x, δ ) = { y ∈ Ω | F Ω ( x, y ) < δ } (the right op en b al l of c enter x and r adius δ ); ◦ B ′ ( x, δ ) = { y ∈ Ω | F Ω ( y , x ) < δ } (the l eft op en b al l of c enter x and r adius δ ); ◦ S ( x, δ ) = { y ∈ Ω | F Ω ( x, y ) = δ } ( the ri ght spher e of c enter x a nd r adius δ ); ◦ S ′ ( x, δ ) = { y ∈ Ω | F Ω ( y , x ) = δ } (the left spher e of c enter x and r adius δ ). In [6] p. 20, H. Busema nn discusses top olog ies fo r general weak metric s paces. In the case of a g enuine metric space, the op en balls are used t o define the top ology of that space. In g e neral, the collections of left and of right open balls in a weak metric space generate tw o different top olo gies. F or the F unk weak metric, we hav e the following If Ω is a b ounded con vex op en set of R n equipp e d with its F unk weak metric; then, the collections of left and of right op en balls ar e sub- bases of the same topo lo gy on 16 A THANASE P AP ADOPO ULOS AND MARC TR OY ANOV Ω, and this top ology coincides with the top o lo gy induced from the inclusion of Ω in R n . In the case where th e co nv e x o pen set Ω is unbounded, the left and the rig ht op en balls of the F unk weak metric ar e alwa ys nonc o mpact. In the nex t prop osition, we study these balls in t he cas e where Ω is b ounded. W e recall that a con vex subset of R n is unbounded if and only if it contains a Euclidean r ay . Prop ositi o n 8.10. L et Ω b e a b ounde d c onvex op en subset o f R n , let x b e a p oint in Ω and let δ b e a nonne gative r e al numb er. Then, (1) The right spher e S ( x, δ ) is c onvex as a su bset of R n , and it is c omp act. F urthermor e, this spher e is the image of ∂ Ω by the Eucli de an ho mothety σ of c enter x and fa ctor (1 − e − δ ) . (2) The left spher e S ′ ( x, δ ) is c onvex as a subset of R n , and it is e qual to the interse ct ion with Ω of the image of ∂ Ω by the Euclide an homothety of c ent er x a nd o f factor ( e δ − 1) , fo l lowe d by the Eu clide an c entr al symmetry of c enter x . The spher e S ′ ( x, δ ) is not ne c essarily c omp act. Pr o of. Let y be a p oint in Ω and let us set, as b efore, a + = R ( x, y ) ∩ ∂ Ω. W e hav e the following equiv alences : y ∈ S ( x, δ ) ⇐ ⇒ log | x − a + | | y − a + | = δ ⇐ ⇒ | x − a + | | y − a + | = e δ , which is e asily seen to b e equiv alent to | y − x | = | x − a + | (1 − e − δ ). F rom this fa c t Prop erty (1) follo ws eas ily . T o pr ove Pr op erty (2), let a − = R ( y , x ) ∩ ∂ Ω. W e ha ve the following equiv a le nces: log | y − a − | | x − a − | = δ ⇐ ⇒ | y − a − | = e δ | x − a − | , which is also equiv alent to | y − x | = ( e δ − 1) | x − a − | . Thu s, y ∈ S ′ ( x, δ ) if and only if y is in the in tersection of Ω with the imag e σ ( ∂ Ω) of ∂ Ω by the Euclidean homothety with center x and of facto r ( e δ − 1), followed by the Euclidean central symmetry of center x . This intersection is convex as a subset of R n but it is not necessarily a co mpact subset of (Ω , F ). Thus, S ′ ( x, δ ) is compact if and o nly if σ ( ∂ Ω) is c ontained in Ω.  W e no te the following “lo cal-implies- g lobal” proper ty o f F unk weak metrics. The meaning of the statement is c le a r, and it follows direc tly from Pr op osition 8.10 (1). Corollary 8.11. We c an r e c onstru ct t he b oundary ∂ Ω of Ω fr om the lo c al ge ometry at a ny p oint of Ω . Corollary 8. 12. L et Ω b e a b ounde d op en strictly c onvex subset of R n . Then, the left and right op en b al ls of Ω ar e ge o desic al ly c onvex with r esp e ct to the F u nk we ak metric F Ω . Pr o of. This follows from Pr op osition 8.10 a nd from Corollary 8.8.  W e also deduce from Prop ositio n 8.1 0 tha t f or any x and x ′ in Ω and for an y tw o po sitive rea l num b er s δ and δ ′ , the right spher es S ( x, δ ) and S ( x ′ , δ ′ ) are homothetic. WEAK FINSLER STRUTURES AND THE FUNK METRIC 17 Thu s, for instance, if Ω is the interior of a Euclidea n sphere (resp ectively , of a Euclidean ellipsoid) in R n , then any right sphere S ( x, δ ) is a Euclidean s pher e (resp ectively , an ellipsoid). Note that the pro o f of Pr op osition 8.10 shows that for a fixed x , any tw o right spheres S ( x, δ ) and S ( x, δ ′ ) ar e homothetic by a Euclidea n homothet y of c ent er x , but that in general, a homothety which s ends a sphere S ( x, δ ) to a sphere S ( x ′ , δ ′ ) do es not necessarily s end the center x of S ( x, δ ) to the cen ter x ′ of S ( x ′ , δ ′ ). O ne can see this fact on the following example: Let Ω b e an open Euclidean disk in R n , and let us tak e x to b e the E uc lide a n center o f that disk. T he n, by symmetry , fo r any δ > 0, the right sphere S ( x, δ ) is a Euclidean sphere who se Euclidean and whose metric centers are bo th at x . No w let x ′ be a point whic h is close to the boundary of Ω. Obviously , the Euclidean homothety that sends ∂ Ω to S ( x ′ , δ ) does not send the cen ter of ∂ Ω to the (F unk-)g eometric cen ter of the sphere S ( x ′ , δ ) (recall that the center of this homo thet y is the point x ). Now taking a comp os ition of tw o homotheties, we obtain a Euclidean homothety that sends the geometric sphere S ( x, δ ) to the geometric spher e S ( x ′ , δ ), and that does not preserve the geometric centers of these spheres. Remark 8.13. T he prop erty for a w eak metric on a subset Ω o f R n that all the right spheres are ho mothetic is a lso sha r ed by the metrics induced by Minko wski weak metrics on R n . References [1] J. C. ´ Alv arez Paiv a & C . Dur´ an, An intr o duction fo Finsler ge ometry , Publicaciones de la Escuela V enezolana de Matematicas (1998). [2] D. Bao, R. L. Bryan t, S. S. Chern & Z. Shen (editors), A sampler of Finsler geo metry , MSRI Publications 50, Cam bridge Uni v ersity Press, 2004. [3] D. Bao, S. S. Che rn & Z. Shen, An intr o duction to Riemann-Finsler ge ometry , Graduate T exts in Mathematics, Spri nger V erlag, 2000. [4] D. Bao, C. Robles & Z. 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Troy anov, Section d e Ma th ´ ema tiques, ´ Ecole Pol y technique F ´ ed ´ erale de La usanne, 1015 Lausanne - Switzerland E-mail addr ess : marc.troyan ov@epfl.ch