Hilbert domains quasi-isometric to normed vector spaces

HILBER T DOMAINS QUASI-ISOMETR IC TO NORMED VECTOR SP A CES BRU NO COLBOIS AND P A TRICK VER OVIC Abstract. W e prov e that a Hilber t domain which is q ua si-isometric to a normed vector space is actually a convex p olyto p e. 1. Intr oduction A Hilb ert domain in R m is a metric space ( C , d C ), where C is an op en b o unde d c onvex set in R m and d C is the distance function o n C — called the Hilb e rt metric — defined as fo llo ws. Giv en tw o distinct p oin ts p and q in C , let a and b b e the interse ction p oints of the straight line defined b y p and q with ∂ C so that p = ( 1 − s ) a + sb and q = (1 − t ) a + tb with 0 < s < t < 1. Then d C ( p, q ) : = 1 2 ln[ a, p, q , b ] , where [ a, p, q , b ] : = 1 − s s × t 1 − t > 1 is the cross ratio o f the 4-tuple of ordered collinear p oin ts ( a, p, q , b ). W e complete the definition b y setting d C ( p, p ) : = 0. a b p q ∂ C The metric space ( C , d C ) th us obtained is a complete non-compact geo des ic metric space whose top ology is the one induc ed b y the canonical top o logy of R m and in whic h the affine o p en segmen ts joining t w o p oints of the b o undary ∂ C are geo desics that a re isometric to ( R , | · | ). F or f ur t her information ab out Hilb ert geometry , w e refer to [4, 5, 9, 11] and the excelle nt in tro duction [15] b y So ci´ e-M ´ ethou. The t wo fundamen tal examples of Hilb ert domains ( C , d C ) in R m corresp ond to the case when C is an ellipsoid, which giv es t he Klein mo del of m -dimensional hyperb o lic geometry ( see for Date : Nov ember 10, 20 18. 2000 Mathematics S ubje ct Classific ation. Primary: global Finsler geometry , Secondary: conv exity . 1 2 BRUNO COLBOI S AN D P A TRICK VERO VI C example [15, first c hapter]), and the case when the closure C is a m -simplex for which there exists a norm k·k C on R m suc h that ( C , d C ) is isometric to the normed v ector space ( R m , k·k C ) (see [8 , pages 1 10–113] or [14, pages 2 2–23]). Muc h has b ee n done to study the similarities b etw een Hilb ert and h yp erb olic geometries (see for example [7 ], [16] or [1]), but little literature deals with the question of kno wing to what extend a Hilb ert geometry is close to that of a normed vec tor space. So let us men tion three results in this latter direction whic h are relev an t fo r our presen t w ork. Theorem 1.1 ([10], Theorem 2) . A Hilb ert dom ain ( C , d C ) in R m is isometric to a norme d ve ctor sp ac e if and only if C is the interior of a m -simple x . Theorem 1.2 ([6], Theorem 3.1) . If C is an op en c onvex p olygonal set in R 2 , then ( C , d C ) is Lipschitz e quivalent to Euclide an plane. Theorem 1.3 ([2], Theorem 1.1. See also [17]) . If C is an op en set in R m whose closur e C is a c onvex p olytop e, then ( C , d C ) is Lipschitz e quivalent to Euclide an m -sp ac e . Recall that a con v ex p olytop e in R m (called a conv ex p olygon when m : = 2) is the con v ex h ull of a finite set of p oints whose affine span is the whole space R m . In light of these three results, it is natural to ask whether the conv erse of Theorem 1.3 — whic h generalizes Theorem 1.2 in higher dimensions — holds. In other w ords, if a Hilb ert domain ( C , d C ) in R m is quasi-isometric to a normed ve ctor space, what can b e said ab o ut C ? Here , b y quasi-isometric we mean the following (see [3]) : Definition 1.1. Given real n um b ers A > 1 and B > 0, a metric space ( S, d ) is said to b e ( A, B )- quasi-isometric t o a nor med v ector space ( V , k·k ) if and only if there exists a map f : S − → V suc h that 1 A d ( p, q ) − B 6 k f ( p ) − f ( q ) k 6 Ad ( p, q ) + B for all p, q ∈ S . W e can no w stat e the result of this pap er whic h asserts that the conv erse of Theorem 1.3 is actually true: Theorem 1.4. If a Hilb ert domain ( C , d C ) in R m is ( A, B ) -quasi-isometric to a norme d ve ctor sp ac e ( V , k·k ) for some r e al c onstants A > 1 and B > 0 , then C is the interior of a c onve x p olytop e. 2. Proof of Theorem 1.4 The pro of of Theorem 1.4 is based o n an ide a dev elop ed b y F¨ ortsc h and Karlsson in t heir pap er [10]. It needs the follo wing fact due to K arlsson and Nosk ov: Theorem 2.1 ([12], Theorem 5.2) . L et ( C , d C ) b e a Hilb ert domai n in R m and x, y ∈ ∂ C such that [ x, y ] 6⊆ ∂ C . Then, given a p oint p 0 ∈ C , ther e exists a c onstant K ( p 0 , x, y ) > 0 s uch that for any se quenc es ( x n ) n ∈ N and ( y n ) n ∈ N in C that c onv e r ge r esp e ctivel y to x and y in R m one c an find an inte g e r n 0 ∈ N for which we have d C ( x n , y n ) > d C ( x n , p 0 ) + d C ( y n , p 0 ) − K ( p 0 , x, y ) for al l n > n 0 . 3 No w, here is the k ey result whic h gives the pro of o f Theorem 1.4: Prop osition 2.1. L et ( C , d C ) b e a Hilb ert doma in in R m which is ( A, B ) - q uasi-isometric to a norme d ve ctor sp ac e ( V , k·k ) for some r e al c onstants A > 1 and B > 0 . Then, if N = N ( A, k·k ) d e notes the ma x imum numb er of p oints in the b al l { v ∈ V | k v k 6 2 A } whose p airwise di s tan c es with r esp e ct to k·k ar e gr e ater than or e q ual to 1 / ( 2 A ) , and if X ⊆ ∂ C is such that [ x, y ] 6⊆ ∂ C for al l x, y ∈ X with x 6 = y , we hav e card( X ) 6 N . Pr o of. Let f : C − → V suc h that (2.1) 1 A d C ( p, q ) − B 6 k f ( p ) − f ( q ) k 6 Ad C ( p, q ) + B for all p, q ∈ C . First o f all, up to translations, w e ma y assume that 0 ∈ C and f (0) = 0. Then suppose t hat there exis ts a subs et X of the b oundary ∂ C such that [ x, y ] 6⊆ ∂ C for all x, y ∈ X with x 6 = y and card( X ) > N + 1. So, pic k N + 1 distinct p oin ts x 1 , . . . , x N +1 in X , and for eac h k ∈ { 1 , . . . , N + 1 } , let γ k : [0 , + ∞ ) − → C b e a geo des ic o f ( C , d C ) that satisfies γ k (0) = 0, lim t → + ∞ γ k ( t ) = x k in R m and d C (0 , γ k ( t )) = t f o r all t > 0. This implies that for all integers n > 1 a nd ev ery k ∈ { 1 , . . . , N + 1 } , w e hav e (2.2)     f ( γ k ( n )) n     6 A + B n from the second inequalit y in Equation 2 .1 with p : = γ k ( n ) a nd q : = 0. On t he other ha nd, Theorem 2.1 yields the existence of some in teger n 0 > 1 suc h that d C ( γ i ( n ) , γ j ( n )) > 2 n − K (0 , x i , x j ) for all inte gers n > n 0 and ev ery i, j ∈ { 1 , . . . , N + 1 } with i 6 = j , and hence (2.3)     f ( γ i ( n )) n − f ( γ j ( n )) n     > 2 A − 1 n  K (0 , x i , x j ) A + B  from the first inequality in Equation 2.1 with p : = γ i ( n ) a nd q : = γ j ( n ). No w, fixing an in teger n > n 0 + AB + max { K (0 , x i , x j ) | i, j ∈ { 1 , . . . , N + 1 }} , w e get     f ( γ k ( n )) n     6 2 A for all k ∈ { 1 , . . . , N + 1 } b y Equation 2.2 to gether with     f ( γ i ( n )) n − f ( γ j ( n )) n     > 1 2 A for all i, j ∈ { 1 , . . . , N + 1 } with i 6 = j by Equation 2.3. But this con tradicts the definition of N = N ( A, k·k ). Therefore, Prop osition 2.1 is pro ve d.  Remark. Giv en v ∈ V suc h that k v k = 2 A , w e ha v e k− v k = 2 A and k v − ( − v ) k = 2 k v k = 4 A > 1 / (2 A ), whic h sho ws that N > 2. The second ingredien t w e will need for the pro of of Theorem 1.4 is the following: 4 BRUNO COLBOI S AN D P A TRICK VERO VI C Prop osition 2.2. L et C b e an op en b ounde d c onvex set in R 2 . If ther e exists a non-empty fin i te subset Y of the b ounda ry ∂ C such that for every x ∈ ∂ C one c an find y ∈ Y w i th [ x, y ] ⊆ ∂ C , then the clos ur e C is a c onvex p olygon. Pr o of. Assume 0 ∈ C and let us consider the con tinuous map π : R − → ∂ C whic h assigns to eac h θ ∈ R the unique in tersection p oin t π ( θ ) of ∂ C with the half-line R ∗ + (cos θ , sin θ ). F or eac h pair ( x 1 , x 2 ) ∈ ∂ C × ∂ C , denote b y A ( x 1 , x 2 ) ⊆ ∂ C the arc segmen t defined by A ( x 1 , x 2 ) : = π ([ θ 1 , θ 2 ]), where θ 1 and θ 2 are the unique real num bers suc h that π ( θ 1 ) = x 1 and π ( θ 2 ) = x 2 with θ 1 ∈ [0 , 2 π ) and θ 1 6 θ 2 < θ 1 + 2 π . Before pro ving Prop osition 2.2 , no tice that adding a p oin t of ∂ C to Y do es not c hange Y ’s prop erty at all, and therefore we ma y assume tha t card( Y ) > 2. So, write Y = { x 1 , . . . , x n } with x 1 = π ( θ 1 ) , . . . , x n = π ( θ n ), where θ 1 ∈ [0 , 2 π ) and θ 1 < · · · < θ n < θ n +1 : = θ 1 + 2 π , a nd let x n +1 : = π ( θ n +1 ) = x 1 . Fix k ∈ { 1 , . . . , n } and pic k an ar bit r a ry x ∈ A ( x k , x k +1 ) r { x k , x k +1 } . By hy p othesis , one can find y ∈ Y with [ x, y ] ⊆ ∂ C . Then the con v ex set C is con tained in one of the tw o op en half-planes in R 2 b ounded b y the line passing through the p oin ts x and y , and hence either A ( x, y ) = [ x, y ], o r A ( y , x ) = [ x, y ]. Since x k ∈ A ( y , x ) and x k +1 ∈ A ( x, y ), w e then hav e x k ∈ [ x, y ] or x k +1 ∈ [ x, y ], whic h yields A ( x k , x ) = [ x k , x ] or A ( x, x k +1 ) = [ x, x k +1 ]. Conclusion: A ( x k , x k +1 ) = S k ∪ S k +1 , where S k : = { x ∈ A ( x k , x k +1 ) | A ( x k , x ) = [ x k , x ] } and S k +1 : = { x ∈ A ( x k , x k +1 ) | A ( x, x k +1 ) = [ x, x k +1 ] } . No w, the set S k (resp. S k +1 ) satisfies [ x k , x ] ⊆ S k (resp. [ x, x k +1 ] ⊆ S k +1 ) whenev er x ∈ S k (resp. x ∈ S k +1 ). So, if w e consider α 0 : = max { θ ∈ [ θ k , θ k +1 ] | A ( x k , π ( θ )) = [ x k , π ( θ )] } , w e hav e S k = [ x k , π ( α 0 )] and S k +1 = [ π ( α 0 ) , x k +1 ]. Hence, A ( x k , x k +1 ) is the union o f the t w o affine segmen ts [ x k , π ( α 0 )] a nd [ π ( α 0 ) , x k +1 ]. Finally , since ∂ C = n [ k =1 A ( x k , x k +1 ), this implies that ∂ C is the union of 2 n a ffine segmen ts in R 2 , and th us C is a con v ex p olygon.  Before pro ving Theorem 1.4, let us recall the following useful result, where a con ve x p olyhe dr on in R m is the in tersection of a finite n umber of closed ha lf-spaces: Theorem 2.2 ([13], Theorem 4.7) . L et P b e a c onvex set in R m and p ∈ ◦ P . Then P is a c onvex p olyhe d r on if and o n ly if al l its pl a ne se ctions c ontaining p ar e c onvex p olyhe dr a. Pr o of of The or em 1.4. Let ( C , d C ) b e a non-empty Hilb ert domain in R m that is ( A, B )-quasi-isometric to a normed v ector space ( V , k·k ) for some real constan ts A > 1 and B > 0. According to Theorem 2.2, it suffices to pro ve Theorem 1.4 for m : = 2 since an y plane section of C giv es rise to a 2-dimensional Hilb ert domain whic h is also ( A, B )-quasi-isometric to ( V , k· k ). So, let m : = 2 , and consider the set E : = { X ⊆ ∂ C | [ x, y ] 6⊆ ∂ C for all x, y ∈ X with x 6 = y } . It is not empt y since { x, y } ∈ E fo r some x, y ∈ ∂ C with x 6 = y (indeed, C is a non-empty op en set in R 2 ), whic h implies together with Prop osition 2.1 that n : = max { card( X ) | X ∈ E } do es exist and satisfies 2 6 n 6 N (recall that N > 2). 5 Then pic k Y ∈ E suc h that card( Y ) = n , write Y = { x 1 , . . . , x n } , and pro ve that for ev ery x ∈ ∂ C o ne can find k ∈ { 1 , . . . , n } such that [ x, x k ] ⊆ ∂ C . Owing to Prop osition 2.2, this will show tha t C is a conv ex p olygon. So, supp ose that there exists x 0 ∈ ∂ C satisfying [ x 0 , x k ] 6⊆ ∂ C for a ll k ∈ { 1 , . . . , n } , and let us find a con tradiction by considering Z : = Y ∪ { x 0 } . First, since x 0 6 = x k for all k ∈ { 1 , . . . , n } (if not, w e w ould g et an index k ∈ { 1 , . . . , n } such that [ x 0 , x k ] = { x 0 } ⊆ ∂ C , whic h is false), w e hav e x 0 6∈ Y . Hence card( Z ) = n + 1. Next, since Y ∈ E and [ x 0 , x k ] 6⊆ ∂ C for all k ∈ { 1 , . . . , n } , w e ha v e Z ∈ E . Therefore, the assumption of the existence of x 0 yields a set Z ∈ E whose cardinality is greater than that of Y , whic h con tradicts the v ery definition of Y . Conclusion: C is a con v ex p olygon, and this prov es Theorem 1.4.  Reference s [1] Benoist, Y. A sur vey on co nv ex divisible sets. Lecture notes, International Conference and Instructional W o rkshop on Discrete Groups in Beijing, 2 006. [2] Bernig, A . Hilb ert geometry of p olytop es. T ech. rep., T o app ear in Ar chiv der Mathematik , 2 008. [3] Burago, D., Burago, Y. , a n d Iv anov, S. A c ourse in metric ge ometry . AMS, 2001 . [4] Busemann, H. The ge ometry of ge o desics . Academic Pr e ss, 1955 . 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Bruno Colbois, Universit ´ e de Neuch ˆ atel, I n s titut de ma th ´ ema tique, Rue ´ Emile Argand 11, Case post al e 158, CH–2009 Neuch ˆ atel, Switzerland E-mail addr ess : bruno.c olbois @unine.ch P a trick Vero vic, UMR 5127 du CNRS & Universit ´ e de Sa voie, Labora toire d e ma th ´ ema tique, Campus scientifique, 73376 Le Bo urge t-du-Lac Cedex, France E-mail addr ess : verovic @univ- savoie.fr