Cyclic projectors and separation theorems in idempotent convex geometry

Cyclic pro jectors and separation theorems in idemp oten t con v ex geometry ∗ St ´ ephane Gaub ert † and Serg e ˘ ı Sergeev ‡ § Abstract Semimo dules o ver idemp oten t semirings like the max-plu s or tropical semiring hav e m uch in common with con v ex cones. This analogy is particularly apparent in the case of subsemimo d ules of th e n -fold cartesian pro duct of the max-plus semiring: it is kno wn that one can separate a ve ctor from a closed subsemimo dule that does n ot con tain it. W e establish here a more general sep aration theorem, wh ic h applies to an y finite col lection of c losed sub semimo dules with a trivial in tersection. In order to pro ve this theorem, we in v estigate the sp ectral prop erties of certain n onlinear op era- tors called her e idemp oten t cyclic p ro jectors. These are idemp oten t analogues of the cyclic nearest-p oin t pr o jections kno wn in con v ex analysis. The sp ectrum of idemp otent cyclic p ro jectors is c haracterized in terms of a suitable extension of Hilb ert’s pr o jec- tiv e metric. W e dedu ce as a corollary of our main r esults the idemp oten t analogue of Helly’s theorem. Keywor ds: Idemp oten t analysis, tropical semiring, semimo dule, con v ex geometry , sep- aration, cyclic pro jections, Hilb ert’s p ro jectiv e metric. AMS classific ation (2000): 52A20 (p rimary), 06F15, 47H07, 52A01 (secondary). 1 In tro duction Some nonlinear pro blems in o ptimization theory a nd mat hematical phys ics turn o ut t o b e linear ov er semirings with an idemp oten t addition ⊕ [1], [8], [14]. W e recall tha t the idemp o- tency of ⊕ means a ⊕ a = a fo r all a , and that the role of this addition is most often pla yed ∗ Suppo rted by the RFBR gr ant 05- 01-00 824 and the joint RFBR/CNRS grant 05-0 1-028 07 † INRIA, Ro cquenco urt, B.P . 105, 78105 Le Chesnay cedex , F r ance. E -mail: Stephane.Ga uber t@inria.fr ‡ Department of Physics, Sub-Department of Quant um Statistics and Field Theory , Mo scow State Uni- versit y , Mo scow, 1199 9 2 Leninskie Gory , Russia. E-mail: sergiej@g mail.com § Corresp o nding a utho r . 1 b y the o p erations of taking maxima o r minima. The searc h for idemp otent analo g ues of classical results has motiv ated the dev elopmen t of idemp oten t mathematics, see the recen t collection of articles [16] and also [15] for more back ground. One of the most studied idempo t en t semirings is t he max-plus semiring. It is the set R ∪ {−∞} equipp ed with the o p erations of a dditio n a ⊕ b := max( a, b ) and multiplication a ⊙ b := a + b . Th e zero elemen t 0 of this semiring is equal to −∞ , and the semiring unity 1 is equal to 0 . So me algebraic structures which coincide with the max-plus semiring (up to isomorphism) ha ve app eared under other na mes. In particular, the min-plus or tropical semiring is obtained by replacing −∞ by + ∞ and max( a, b ) b y min( a, b ) a b o v e. Applying x 7→ exp( x ) to the max-plus semiring (assume exp( −∞ ) = 0 ), we obtain the max-t imes semiring, further denoted by R max , m . It is the set o f nonnegative num b ers ( R + ), equipped with the op eratio ns a ⊕ b = max( a, b ) and a ⊙ b = a × b . The zero and unit elemen ts of R max , m coincide with the usual 0 and 1. Our main results (Sect. 4) will b e stated o v er this semiring, as it mak es clearer some analo g ies with classical con v ex a na lysis. W e shall consider here subsemimo dules o f the n -fold cartesian pro duct K n of a semiring K and, more generally , of the set K I of functions from a set I to K . F urther examples can b e found e.g. in [1], [14] and [17]. In a n idemp otent semiring, there is a canonical o rder relation, for whic h ev ery elemen t is “nonnegativ e”. Therefore, idemp otent semimodules ha v e m uc h in common with the semi- mo dules ov er the semiring of no nnegativ e n umbers, that is, with c onvex c ones [19]. One o f the first results based on this idea is the separation theorem for con v ex sets o v er “extremal algebras” prov ed b y K. Zimmermann in [21]. This theorem implies that a p oint in R n max , m , whic h do es not b elong to a semimo dule that is closed in the Euclidean top ology , can b e separated from it by an idemp o ten t analogue of a closed halfspace. Generalizations o f this result w ere obtained in a work b y S.N. Samborski ˘ ı and G.B. Shpiz [20 ] and in w orks by G. Cohen, J.-P . Quadrat, I. Singer, and the first author [5], [6]. In the sp ecial case of finitely generated semimo dules, a separation theorem ha s a lso b een obtained b y M. Dev elin a nd B. Sturmfels in [10], with a strong emphasis on some com binatoria l asp ects of the result. The main result of t his pap er, Theorem 21, sho ws that sev e r al closed semimo dules whic h do not ha v e common nonzero p oin ts can b e separated fro m eac h other. This means that for eac h of these semimo dules, w e can select a n idemp oten t halfspace con taining it, in suc h a w a y tha t these half spaces a lso do not ha v e common nonzero p oin ts. Ev en in the case of tw o semimodules, this statemen t has not b een pro v ed in the idemp oten t literature. Inde ed, the earlier separation theorems deal with the separation of a p oint from an (idempo t ent) con v ex set or semimo dule, rather t han with the separation o f t w o conv ex 2 sets or semimo dules. Not e that unlik e in the classical case, separating tw o con ve x sets cannot b e reduced to separating a p oin t from a con v ex set. More precisely , it is easily sho wn that t w o con v ex sets A and B can b e separated if and only if the p oint 0 can b e separated from their Minko wski difference A − B , in classical con v ex geometry . In idemp o ten t geometry , an analogue of Mink owski difference can still b e defined, consider A ⊖ B = { x | ∃ b ∈ B : x ⊕ b ∈ A } . Ho w ev er, due t o the idemp otency of the addition, we cannot recov er a halfspace separating A and B from a halfspace separating 0 from A ⊖ B . In order to prov e the main result, Th eorem 21, w e inv estigate the sp ectral prop erties of idemp oten t cyclic pro jectors. By idemp ot ent cyclic pro jectors we mean finite comp ositions of certain nonlinear pr o jectors on idemp oten t semimo dules. The contin uity and homog eneity of these nonlinear pro j ectors enables us to apply to their compo sitions, i.e. to the cyclic pro- jectors, some results from non-linear P erron-F rob enius theory . The main idea is to pro v e the equiv alence of the follow ing three statemen ts: 1 ) that the semimo dules ha v e trivial in tersec- tion, 2) that the separating half spaces exist, and 3 ) that the sp ectral radius of the asso ciated cyclic pro jector is strictly less than 1 . This equiv a lence is established in Theorems 16 and 19, whic h deal with the sp ecial case of archimede an semimodules, i.e. semimodules con t aining at least one p ositiv e v ector. As a n ingredien t o f the pro of, w e use a nonlinear extension o f Collatz-Wielandt’s theorem obtained by R.D. Nussbaum [18]. T o deriv e the main separatio n result, Theorem 2 1, we sho w that for any collection of trivially in tersecting semimo dules, there is a collection of trivially intersec ting ar chime de an semimo dules, suc h that ev ery semi- mo dule from the first collection is con tained in an arc himedean semimo dule from the second collection. W e also sho w in Theorems 13 and 15 tha t the orbit of a n eigen v ector of a cyclic pro jector maximizes a certain ob jectiv e function. W e call this maximum the Hilb ert v alue of semi- mo dules, as it is a natural generalization of Hilb ert’s pro jectiv e metric, and characterize the sp ectrum of cyclic pro jectors in terms of these Hilb ert v alues (Theorem 18). The pro jectors on idemp o t ent semimo dules, whic h constitute the cyclic pro j ectors considered here, ha ve b een studied by R.A. Cuninghame-Green, see [7] and [8], Chapter 8, where they app ear as AA ∗ -pro ducts. The g eometrical prop erties of these pro jectors ha v e b een used in [5, 6] to establish separation theorems. The same op erat o rs hav e also b een studied by G.L. Litvinov, V.P . Maslo v and G.B. Shpiz, who obtained in [17] idemp otent analog ues of sev eral r esults from functional analysis, including the analytic f o rm of the Hahn-Banach theorem. The idemp oten t cyclic pro j ectors ha ve been introduced, in the case of t w o semimo dules, by P . Butko vi ˇ c and R .A. Cuninghame-Green in [9], where these op erators giv e rise to an efficien t 3 (pseudo-p olynomial) algorithm for finding a p oint in the interse ction of tw o finitely generated subsemimodules of R n max , m . In conv ex analysis and o pt imizatio n theory , an analogous role is pla y ed by the cyclic nearest-po in t pro jections on con v ex sets [3]. As a corolla ry of Theorems 19 and 21, we deduce a max-plus analogue of Helly’s theorem. This r esult has also b een obtained, with a differen t pro of, b y F. Meunier and the first a uthor [13]. Our main results apply to subsemim o dules of R n max , m . Some of our results still hold in a more general setting, see Sect. 3. Ho w ev er, the separation of sev eral semimo dules in suc h a generalit y remains an op en question. The results of this pap er ar e presen ted as follow s. Sect. 2 describes t he main assumptions that are satisfied by the semimo dules of the pap er. Besides that, it is occupied by some basic notions and f a cts that will b e used further. Sect. 3 is dev oted to the r esults obtained in the most general setting, with resp ect to the assumptions of Sect. 2. The main results for the case R n max , m are o btained in Sect. 4. These results include separation o f se ve ral semimo dules and c haracterization of the sp ectrum of cyclic pro jectors. 2 Preliminary results o n pro jector s and separatio n W e start this section with some details concerning the role of partia l order in idemp o t ent algebraic structures. F o r more background, w e refer the reader to e.g. [1, 8, 17]. The idemp otent addition ⊕ defines the canonical order relation ≤ ⊕ on the semiring K b y the rule λ ⊕ µ = µ ⇔ λ ≤ ⊕ µ for λ, µ ∈ K . The idemp oten t sum λ ⊕ µ is equal to the least upp er b o und sup( λ, µ ) with resp ect to the or der ≤ ⊕ . The idemp oten t sum of an arbitrary subset is defined to b e the least upp er b ound of this subset, if this least upper b ound exists. In a semimo dule V , w e define the or der relation ≤ ⊕ , V in the same w a y . The relation λ ≤ ⊕ µ b et w een λ, µ ∈ K implies λx ≤ ⊕ , V µx f o r all x ∈ V . When V = K n and K = R max , m , the order ≤ ⊕ coincides with the usual linear order on R + , and the order ≤ ⊕ , V coincides with the standard p oint wise order on R n . F o r this r eason, we will write ≤ instead o f ≤ ⊕ and ≤ ⊕ , V , in the sequel. A semiring or a semimo dule will b e called b -c omplete (see [17]), if it is closed under the sum (i.e. the suprem um) of an y subset b ounded fro m ab o v e, a nd the m ultiplication distributes o v er suc h sums. If the least upp er b ound ⊕ exists for all subsets b o unded from ab o v e, then the g reatest lo w er b ound ∧ exists for all subsets b ounded from b elow . Consequen tly , the 4 greatest low er b o und exists for any subset of a b -complete semiring or a semimo dule, since suc h a subset is b o unded from b elow b y 0 . Also no te t hat if K is a b -complete semiring, and the set K \ { 0 } is a m ultiplicativ e group, then this group is a b elian by Iw asaw a’s theorem [2]. A semiring K suc h that the set K \ { 0 } is an ab elian m ultiplicativ e gro up is called a n idemp o tent semifield . W e shall consider semirings K and semimodules V ov er K that satisfy the following assump- tions: ( A 0): the semiring K is a b -complete idem p o ten t se mifield, and the semimo dule V is a b -complete semimo dule ov er K ; ( A 1): fo r all elemen ts x a nd y 6 = 0 from V , the set { λ ∈ K | λy ≤ x } is b ounded from ab o ve . Assumptions ( A 0 , A 1) imply that the op eration x/y = max { λ ∈ K | λy ≤ x } . (1) is defined for all elemen ts x and y 6 = 0 f rom V . The follo wing can b e view ed as another definition of the op eratio n / equiv alent to (1): λy ≤ x ⇔ λ ≤ x/y . (2) In the case V = K I , x/y = ^ i : y i 6 = 0 x i /y i . (3) The op eration / has the following prop erties: ( ^ α x α ) /y = ^ α ( x α /y ) , ( x/ M α y α ) = ^ α ( x/y α ) (4) ( λx ) /y = λ ( x/y ) ∀ λ, y / ( λx ) = λ − 1 ( y /x ) ∀ λ 6 = 0 . (5) W e also need the follo wing lemma. Lemma 1 Under ( A 0 , A 1) , x/x = 1 for al l nonzer o ve ctors x ∈ V . If λx = x for a nonzer o ve ctor x ∈ V , then λ = 1 . Pr o of . The inequality x ≤ x implies that x/x ≥ 1 , see (1 ). On the o ther hand, w e hav e that ( x/x ) x ≤ x . Multiplying this b y x/x , we obtain that ( x/x ) 2 x ≤ ( x/x ) x ≤ x , hence ( x/x ) 2 ≤ x/x and x/x ≤ 1 . Thus x/x = 1 . If λx = x for some x 6 = 0 , then λ ( x/x ) = ( λx ) /x = x/x and so λ = 1 . 5 Definition 2 A s ubse mimo dule V of V is a b -(sub)semimo dule , if V is cl o se d under the sum of an y of its subsets b ounde d fr om ab ove i n V . Let V b e a b -subsemim o dule o f the semimo dule V . Consider t he o p erator P V defined b y P V ( x ) = max { u ∈ V | u ≤ x } , (6) for ev ery elemen t x ∈ V . Here we use “max” to indicate that the least upp er b o und b elongs to the set. The op erator P V is a pr oje ctor on t o the subsemimo dule V , as P V ( x ) ∈ V for any x ∈ V and P V ( v ) ∈ V for an y v ∈ V . In principle, P V can b e defined for all subsets of V , if w e write sup instead of max in (6), but then P V ma y not b e a pro jector on V . Definition 3 A s ubs e m imo dule V o f V is c al le d elemen ta ry , if V = { λy | λ ∈ K} for some y ∈ V . The pr oje ctor onto such a se m imo dule i s also c al le d elemen tary . Assumptions ( A 0 , A 1) imply that elemen tary semimo dules are b -semimo dules. F or the ele- men tary semimo dule V = { λy | λ ∈ K} , the pro jector P V is g iven b y P V ( x ) = ( x/y ) y , and this fact can b e generalized as follows . Prop osition 4 If V is a b -subsemimo dule of V and P V ( x ) = λy for some λ ∈ K and x, y ∈ V , then P V ( x ) = ( x/y ) y . Pr o of . If V is a b -semimo dule, then y ∈ V , and ( x/y ) y ≤ x implies tha t ( x/y ) y ≤ P V ( x ) = λy . On the other hand, λy ≤ x implies λ ≤ x/y and λy ≤ ( x/y ) y . Note that P V is isotone with resp ect to inclusion: U ⊂ V ⇒ P U ( x ) ≤ P V ( x ) for all x. (7) It is also homogeneous and isotone: P V ( λx ) = λP V ( x ) , x ≤ y ⇒ P V ( x ) ≤ P V ( y ) . (8) W e remark that the o p erator P V is in general not linear with resp ect to ⊕ o r ∧ op erations, ev en in the case V = R n max , m . In idemp oten t g eometry , the role of halfspace is play ed by the following ob j ect. Definition 5 A set H given by H = { x | u /x ≥ v /x } ∪ { 0 } (9) with u, v ∈ R n max , m , u ≤ v , wil l b e c al le d ( idemp o ten t) halfspace . 6 Prop erties (4) and (5) of the op eration / imply that any halfspace is a semimo dule. If V = K I , then w e can use (3) and then H = { x | ^ i : x i 6 = 0 u i x − 1 i ≥ ^ i : x i 6 = 0 v i x − 1 i } ∪ { 0 } . (10) If V = K n and all co ordinates of u and v ar e no nzero, t hen we ha v e that H = { x | M 1 ,...,n x i u − 1 i ≤ M 1 ,...,n x i v − 1 i } . (11) Suc h ide mp ot en t halfspaces fo rmally resem ble the closed homogeneous halfspaces of the finite-dimensional con v ex geometry [19]. Since the op eration / is isotone with resp ect to t he first a rgumen t, w e can replace the inequalities in (5), (10) and (11) by the equalities. F or instance, definition (5 ) can b e rewritten as H = { x | u /x = v /x } ∪ { 0 } , (12) where u ≤ v . The presen t pap er is concerned with the separation of sev eral b -semimodules, whereas the separation theorems whic h hav e b een established previously , like the ones of [5, 6], deal with the separation of one p oin t from a semimo dule. F or the conv enience of the reader, we next state a theorem, whic h is a v ariant of a separatio n theorem of [5]. The difference is in that w e deal with b -complete semimo dules rather tha n with complete semimodules. Both results are closely related with the idemp oten t Hahn-Banac h theorem of [17]. Theorem 6 (Compare with [5], Theorem 8) L e t V b e a b -subsem imo dule of V a n d let u / ∈ V . Then the h alfsp ac e H = { x | P V ( u ) /x ≥ u/x } ∪ { 0 } (13) c ontains V but n ot u . Pr o of . T ake a nonzero v ector x ∈ V (the case x = 0 is trivial). Since ( u/x ) x ≤ u , w e hav e ( u/x ) x ≤ P V ( u ), whic h is b y (2) equiv alent to u/x ≤ P V ( u ) /x . Hence V ⊆ H . T ak e x = u and assume that P V ( u ) /u ≥ u/u = 1 . This is equiv alen t to u ≤ P V ( u ) and hence to u = P V ( u ). Since V is a b -semimo dule, we ha v e that u ∈ V , whic h is a con tradiction. Hence u / ∈ H . Definition 7 Consider the pr e or der r elation  define d b y x  y ⇔ y /x > 0 . (14) 7 We say that x and y ar e comparable , and we write x ∼ y , if x  y an d y  x . Equivalently, x ∼ y ⇔ ( x/y )( y /x ) > 0 . (15) Note that if y = λx with λ 6 = 0 , then y ∼ x , and that the inequalit y x ≤ y , if x 6 = 0 , implies that x  y . In particular, P V ( x )  x for an y nonzero x ∈ V and any semimodule V , pro vided that P V ( x ) is nonzero. When V = K n , comparability can b e characterized in terms of suppo rts. Recall that the supp ort of a v ector x in K n is defined by supp( x ) = { i | x i 6 = 0 } . It can b e c hec k ed tha t for all x, y ∈ K n , w e ha v e x  y iff supp( x ) ⊂ supp( y ), and so, x ∼ y iff supp( x ) = supp ( y ). Prop osition 8 L et x ∈ V b e a nonz e r o ve ctor and let V ⊆ V b e a b -semimo dule c ontaining a nonz e r o ve ctor y . If y  x , then P V ( x ) is nonzer o, and y  P V ( x )  x . If y ∼ x , then P V ( x ) ∼ x . Pr o of . By the definition of / and by (14), there exists α suc h that α y ≤ x . Then αy ≤ P V ( x ), hence P V ( x ) is nonzero and y  P V ( x ). Prop osition 9 L et F b e an isotone and ho mo gene ous op er ator, let λ, µ b e arbitr ary sc alars fr om K and let v and u b e nonzer o ve ctors such that v ≺ u . Supp ose that one of the fol lowing is true: 1. F v ≥ µv and F u = λu ; 2. F v = µv and F u ≤ λu . Then µ ≤ λ . Pr o of . Applying F to the inequalit y ( u/v ) v ≤ u and using an y of the g iv en conditions, w e o btain that ( u/v ) µ v ≤ λu . If λ = 0 , then µ = 0 . If λ is inv ertible, then b y (2) ( u/v ) µλ − 1 ≤ u/ v . Cancelling u/v , w e get µ ≤ λ . Prop erties (4) and ( 5 ) imply that the sets { x | x  y } , { x | x  y } and hence { x | x ∼ y } are subsemimodules of V . F or an y semimo dule V ⊂ V and any v ector y ∈ V , w e define V y = { x ∈ V | x  y } , (16) whic h is a subsemimodule of V . When V = K n , V y is uniquely determined b y the supp ort M o f y . F or this reason, for all M ⊆ { 1 , . . . , n } , w e set V M = { x ∈ V | supp( x ) ⊂ M } . (17) 8 Definition 10 A ve ctor x ∈ V is c al le d arc himedean , if y  x for al l y ∈ V . A semim o dule V ⊆ V is c al le d arc himedean , if i t c ontains an ar chime de an ve ctor. A ha lfsp ac e wil l b e c al le d arc himedean , if b o th ve ctors defining it (e.g. u and v in (9)) ar e a r chime de an. Of course, Def. 10 mak es sense only in the case when V satisfies the following assumption: ( A 2): the semimo dule V has an ar chimede an v ector. This a ssumption is satisfied b y t he semimo dules V = K n (w e are also assuming ( A 0 , A 1)). In this case, arc himedean halfspaces hav e b een written explicitly in (11). 3 Cyclic pro jectors and separation theorems: general results In this section w e study cyclic pro jectors, that is, comp ositions of pro jectors P V k · · · P V 1 , (18) where V 1 , . . . , V k are b -subsemimodules of V . W e assume ( A 0 , A 1), whic h means in particular that K is an idempotent semifie ld, and state general results concerning cyc lic pro jectors and separation prop erties. F or the nota t io nal conv enience, w e will write P t instead of P V t . W e will also adopt a con v ention of cyclic n um b ering of indices of pro jectors and semimo dules, so that P l + k = P l and V l + k = V l for all l . Definition 11 L et x 1 , . . . , x k b e nonzer o elemen ts of V . The value d H ( x 1 , . . . , x k ) = ( x 1 /x 2 ) ( x 2 /x 3 ) . . . ( x k /x 1 ) . (19) wil l b e c a l le d the Hilb ert v a lue of x 1 , . . . , x k . It follows from Def. 7 that d H ( x 1 , . . . , x k ) 6 = 0 if and only if all v ectors x 1 , . . . , x k are com- parable. One can sho w that d H ( x 1 , . . . , x k ) ≤ 1 . This inequalit y is an equalit y if and only if x 1 , . . . , x k differ from eac h other only b y scalar m ultiples. The Hilb ert v alue is in v ar ia n t un- der m ultiplication of an y of its a r g umen ts by an in vertible scalar, and b y cyclic p erm utation of its argumen ts. The Hilb ert v alue of t w o v ectors x 1 , x 2 w as studied in [5]. F o r t w o comparable vec tors in R n max , m , that is, for tw o v ectors with common supp o r t M it is giv en by d H ( x 1 , x 2 ) = min i,j ∈ M ( x 1 i ( x 2 i ) − 1 x 2 j ( x 1 j ) − 1 ) , (20) 9 so that − log( d H ( x 1 , x 2 )) coincides with Hilb ert’s pro jectiv e metric δ H ( x 1 , x 2 ) = log( max i,j ∈ M ( x 1 i ( x 2 i ) − 1 x 2 j ( x 1 j ) − 1 )) = − log( d H ( x 1 , x 2 )) . (21) Definition 12 The Hilb ert v alue of k subsemimo dules V 1 , . . . , V k of V is defi ne d by d H ( V 1 , . . . , V k ) = sup x 1 ∈ V 1 ,...,x k ∈ V k d H ( x 1 , . . . , x k ) (22) Theorem 13 Supp ose that the op er ator P k · · · P 1 has an eigenve c tor y with eigenv alue λ . Then λ = max x 1 ∈ V y 1 ,...,x k ∈ V y k d H ( x 1 , . . . , x k ) = d H ( ¯ x 1 , . . . , ¯ x k ) , (23) wher e ¯ x i = P i · · · P 1 y . Pr o of . Note that ¯ x i , for a n y i , is an eigenv ector of P i + k · · · P i +1 and that a ll these ve ctors are comparable with y . F urther, let x 1 , . . . , x k b e arbitrary elemen ts of V y 1 , . . . , V y k , resp ectiv ely , and let α 1 , . . . , α k b e scalars suc h that α 1 x 2 ≤ P 2 x 1 , . . . α k − 1 x k ≤ P k x k − 1 , α k x 1 ≤ P 1 x k , (24) T ak e the last inequalit y . Applying P 2 to b o th sides and using the first inequalit y , we ha v e that α 1 α k x 2 ≤ P 2 P 1 x k . F urther, w e apply P 3 to this inequality and use the ineq uality α 2 x 3 ≤ P 3 x 2 . Pro ceeding in the same manner, we finally obtain α 1 . . . α k x k ≤ P k · · · P 1 x k . (25) It f o llo ws f rom Prop. 9 that α 1 · · · α k ≤ λ . W e tak e α i = x i /x i +1 for i = 1 , . . . , k − 1, and α k = x k /x 1 . This leads to d H ( V y 1 , . . . , V k y ) ≤ λ. (26) Note tha t this inequalit y is true if V 1 , . . . , V k are not b -semimo dules. Applying Prop. 4 w e ha v e tha t λy = d H ( ¯ x 1 , . . . , ¯ x k ) y . By Lemma 1 w e can cancel y , and the observ ation that ¯ x i ∈ V y i for all i yields the desired equalit y . The situation when P k · · · P 1 has an eigenv ector with nonzero eigen v alue o ccurs, if at least one of the semimo dules V 1 , . . . , V k is eleme ntary , that is, generated b y a single v ector x i , and if all other semimo dules ha v e v ectors comparable with x i . In t his case P k · · · P i +1 x i is the only eigen v ector of P k · · · P 1 with nonzero eigen v alue. T o obtain the follow ing lemma, w e use Prop. 8. 10 Lemma 14 L et x 1 ∈ V 1 and x i = P i x i − 1 for i = 2 , . . . , k . Then, the Hilb ert value d H ( x 1 , . . . , x k ) is not e qual to 0 if and only if V 2 , . . . , V k have v e ctors c omp ar able with x 1 . Theorem 15 Supp ose that the ve ctors x i , i = 1 , . . . ar e such that x 1 ∈ V 1 and x i = P i x i − 1 for i = 2 , . . . . Then d H ( x l +1 , . . . , x l + k ) is nonde cr e asing with l so that the fol lowing in e quali- ties hold fo r al l l : d H ( x 1 , . . . , x k ) ≤ d H ( x l +1 , . . . , x l + k ) ≤ 1 . (27) Pr o of . As V i are b -semimodules, x i ∈ V i for all i . If the Hilb ert v alue is 0 for all l , then there is nothing to pr ov e. So w e assume that there exists a least l = l min for whic h the Hilb ert v alue d H ( x l , . . . , x l + k − 1 ) is nonzero. As it is nonzero, b y Lemma 1 4 , all x l , . . . , x l + k − 1 are comparable. By Prop. 8 , x l + k is also comparable with t hem, and the same is true ab out the rest of the sequence , hence d H ( x l , . . . , x l + k − 1 ) is nonzero for all l ≥ l min . Now w e tak e any l ≥ l min and consider the comp osition P l + k P ′ l + k − 1 · · · P ′ l +1 , (28) where P ′ i , for i = l + 1 , . . . , l + k − 1, are elemen t a ry pro jectors o nto the semimo dules generated b y x i . The op erator (28) has an eigen v ector x l + k . By Theorem 13 d H ( x l , . . . , x l + k − 1 ) ≤ max y ∈ V l , y  x l + k d H ( x l +1 , . . . , x l + k − 1 , y ) = d H ( x l +1 , . . . , x l + k ) . (29) for all l = 1 , . . . . The follow ing theorem assumes the existence of arc himedean v ectors ( A 2 ) . Theorem 16 Supp ose that P k · · · P 1 has an ar ch ime de an eigenve ctor y with nonzer o eig e n- value λ . The fo l lowing ar e e quivalen t: 1. ther e exist a n ar chime de an ve ctor x and a sc ala r µ < 1 such that P k · · · P 1 x ≤ µx ; 2. for al l i = 1 , . . . , k ther e exist ar chime de an halfsp ac es H i such that V i ⊆ H i and H 1 ∩ · · · ∩ H k = { 0 } ; 3. V 1 ∩ · · · ∩ V k = { 0 } ; 4. λ < 1 . Pr o of . 1 ⇒ 2: Denote x 0 = x and x i = P i · · · P 1 x 0 . Note that all the x i are also arc himedean b y Prop. 8. F or all i = 1 , . . . , k w e hav e t ha t V i ⊆ { u : x i − 1 /u = x i /u } = H i . (30) 11 Indeed, if x i − 1 = x i , then H i coincides with the whole V . If x i − 1 6 = x i , whic h means that x i / ∈ V i − 1 , then the inclusion in (30) fo llo ws from Theorem 6. Assume that there exists a nonzero v ector u whic h b elongs to ev ery H i . Then x k /u = x/u . But x k /u ≤ ( µx ) /u ≤ x/u , hence µ ( x/u ) = ( µ x ) /u = x/u . Cancelling x/u , w e g et µ = 1 whic h contradicts 1 . The implication is prov ed. 2 ⇒ 3: Immediate. 3 ⇒ 4: By the conditions of this theorem, P k · · · P 1 has an eigenv ector y with eigenv alue λ . As a ny v ector is greater tha n or equal to its image b y the pro jector P i , we hav e that λ ≤ 1 . Assume that λ = 1 . Then the inequalities P k · · · P 1 y ≤ P k − 1 · · · P 1 y ≤ . . . ≤ y (31) turn in to equalities, and y is a common v ector of V 1 , . . . , V k , whic h contradicts 3 . 4 ⇒ 1: T ak e x = y . T o illustrate Theorem 16, consider the matrices A =    0 0 0 −∞ 1 2 −∞ 1 0 − 1 2 3    , B =    3 2 2 0 0 0 −∞ 0 − 1    . (32) Let a i and b i denote the i - th column of A and B , resp ectiv ely . F or a ll v ectors x = ( x 1 , . . . , x n ) and β > 0 , w e denote by exp( β x ) the v ector of the same size with en tries exp( β x j ). W e define V 1 (resp. V 2 ) to b e t he subsemimodule o f R 3 max , m generated by the vec tors exp( β a i ) for 1 ≤ i ≤ 4 (resp. exp( β b i ) for 1 ≤ i ≤ 3). The discus sions which follo w are indep enden t of the c hoice of the scaling par a meter β > 0, whic h is adjusted to mak e Figure 1 readable ( we to ok β = 2 / 3). The tw o semimo dules V 1 , V 2 and their generators are represen ted as follow s at the left of the figure. Here, a non- zero vec tor w = ( w 1 , w 2 , w 3 ) ∈ R 3 max , m is represen ted b y the p o in t of the t wo dimensional simplex whic h is the barycen ter with w eights w j of the three v ertices o f this simplex. The generators a i and b i corresp ond to the b old dots. The semimo dules V 1 and V 2 corresp ond to the tw o medium gr ey regions, together with the b old brok en segmen ts joining the generators to eac h of these regions. Since the en tries of x 0 := b 2 = exp ( β (2 , 0 , 0)) ∈ V 2 are nonzero, the v ector x 0 is archimed ean, and one can chec k, using the explicit form ula of the pro j ector (Theorem 5 of [5]), that x 0 is an eigen v ector o f P 2 P 1 . Indeed, x 1 := P 1 x 0 = ex p ( β ( − 1 , 0 , 0) T ) (33) 12 and x 2 := P 2 x 1 = ex p ( β ( − 1 , − 3 , − 3) T ) = exp( − 3 β ) x 0 . (34) The halfspaces constructed in the pro of o f Theorem 16 are giv en b y H 1 = { u | x 0 /u = x 1 /u } = { u | min (exp(2 β ) /u 1 , 1 /u 2 , 1 /u 3 ) = min(exp( − β ) / u 1 , 1 /u 2 , 1 /u 3 ) } = { u | max ( u 2 , u 3 ) ≥ exp( β ) u 1 } (35) and H 2 = { u | x 1 /u = x 2 /u } = { u | min(exp( − β ) /u 1 , 1 /u 2 , 1 /u 3 ) = min(exp( − β ) / u 1 , exp( − 3 β ) / u 2 , exp( − 3 β ) / u 3 ) } = { u | u 1 ≥ exp(2 β ) max( u 2 , u 3 ) } . (36) These t w o ha lfspaces are represen ted b y the zones in light gray (righ t). The pro of of Theo- rem 16 sho ws that their inters ection is zero (meaning that it is reduced to the zero v ector). x 1 x 2 x 3 a 3 a 4 b 3 b 2 = x 0 V 2 V 1 x 1 b 1 a 2 a 1 x 1 x 2 x 3 H 2 H 1 Figure 1: Tw o semimo dules (left) with separating halfspaces (right) 4 Cyclic pro jecto rs and separation theo rems in R n max , m In R n max , m , it is natural to consider semimo dules that are closed in the Euclidean top olo g y . One can easily sho w that suc h semimo dules a re b -semimo dules. Theorem 3.11 of [6] implies that pro jectors onto closed subsemimo dules of R n max , m are con tin uous. In order to relax the assumption concerning a rc himedean vec tors in Theorem 16, we shall use some results fro m nonlinear sp ectral theory , that w e next recall. By Bro u w er’s fixed p oin t theorem, a con tin uous ho mo g eneous o p erator x 7→ F x tha t maps R n + to itself has a 13 nonzero eigen v ector. This allows us to define the nonlinear sp ectral radius of F , ρ ( F ) = max { λ ∈ R + | ∃ x ∈ ( R n + ) \ 0 , F x = λx } . (37) Supp ose in addition that F is isotone. Then it can b e c hec k ed that if F x = λx , F y = µy , and if x and y are comparable, then λ = µ . It follows that the num b er of eigen v alues of F is b ounded b y the n um b er of nonempty supp orts of ve ctors of R n + , i.e. b y 2 n − 1 . This implies in particular that the maxim um is attained in (37). W e shall need the following nonlin- ear generalization of the Collatz-Wielandt fo r m ula for t he sp ectral radius of a nonnegativ e matrix. Theorem 17 (R.D. Nussbaum, Theorem 3.1 of [18]) F or any isotone, homo gene ous, and c ontinuous map F fr om R n + to itself, we ha v e: ρ ( F ) = inf x ∈ ( R + \{ 0 } ) n max 1 ≤ i ≤ n [ F ( x )] i x − 1 i . (38) This result implies that the sp ectral ra dius of suc h op erato r s is isotone: if F ( x ) ≤ G ( x ) for an y x ∈ R n + , then ρ ( F ) ≤ ρ ( G ). As the pro jectors on subsemimo dules of R n max , m are isotone, homogeneous and contin uous, so are their comp o sitions, i.e. cyclic pro jectors. Consequen tly , we can apply Theorem 1 7 to them. Hence, if V ′ i , i = 1 , . . . , k and V i , i = 1 , . . . , k are closed semimo dules in R n max , m and suc h that V ′ i ⊆ V i , i = 1 , . . . , k , then ρ ( P ′ k · · · P ′ 1 ) ≤ ρ ( P k · · · P 1 ) , (39 ) as the pro jectors are isotone with resp ect t o inclusion ( 7). In the follo wing theorem w e c haracterize the sp ectrum o f cyclic pro jectors in R n max , m . Theorem 18 L et V 1 , . . . , V k b e clos e d semim o dules in R n + . Then the Hilb ert value d H ( V 1 , . . . , V k ) is the sp e c tr al r adius of P k · · · P 1 . The s p e ctrum of P k · · · P 1 is the set o f Hilb ert v a lues d H ( V M 1 , . . . , V M k ) , wher e M r ange s over al l non empty subsets of { 1 , . . . , n } . Pr o of . W e first prov e that the Hilb ert v alue d H ( V 1 , . . . , V k ) is the sp ectral radius of the cyclic pro jector, and hence an eigen v alue. W e tak e k elemen tary subsemimo dules spanned b y x i ∈ V i , i = 1 , . . . , k and consider elemen ta r y pro jectors P ′ i on to them. Observ e that ρ ( P ′ k · · · P ′ 1 ) = d H ( x 1 , . . . , x k ) . (40) Denote b y ¯ x 0 an eigen vec tor of P k · · · P 1 , asso ciated with the sp ectral radius, and let ¯ x i = P i · · · P 1 ¯ x 0 . Then ρ ( P k · · · P 1 ) = d H ( ¯ x 1 , . . . , ¯ x k ) . (41) 14 By (39) it follo ws that ρ ( P k · · · P 1 ) ≥ ρ ( P ′ k · · · P ′ 1 ), that is, d H ( ¯ x 1 , . . . , ¯ x k ) ≥ d H ( x 1 , . . . x k ) for an y x 1 ∈ V 1 , . . . , x k ∈ V k . Th us, the Hilb ert v alue of V 1 , . . . V k is the spectral radius of P k · · · P 1 . No w consider d H ( V M 1 , . . . , V M k ) for arbitrary M ⊆ { 1 , . . . , n } . Note that the semimo dules V M 1 , . . . , V M k are closed, and denote by P M 1 , . . . , P M k the pro jectors on to these. It is easy to see that P M i ( y ) = P i ( y ) f o r all i and all y with supp ( y ) ⊆ M . It follows that d H ( V M 1 , . . . , V M k ) is the sp ectral radius of P M k · · · P M 1 and also an eigen v alue o f P k · · · P 1 . W e hav e prov ed that a n y Hilb ert v alue d H ( V M 1 , . . . , V M k ) is an eigen v alue of P k · · · P 1 . The con v erse statement follo ws from Theorem 13. The follow ing three results refine Theorem 16. Theorem 19 Supp ose that V 1 , . . . , V k ar e c l o se d ar chime de an subsem i m o dules of R n max , m . The fo l lowing ar e e quivale n t: 1. ther e exist a p ositive ve ctor x and a numb er λ < 1 such that P k · · · P 1 x ≤ λx ; 2. ther e exis t ar chime de an halfs p ac e s H i which c o n tain V i and ar e such that H 1 ∩ · · · ∩ H k = { 0 } ; 3. V 1 ∩ · · · ∩ V k = { 0 } ; 4. ρ ( P k · · · P 1 ) < 1 . Pr o of . The implications 1 ⇒ 2, 2 ⇒ 3 3 ⇒ 4 are pro v ed in Theorem 16. The implication 4 ⇒ 1 follo ws from Equation (38). Prop osition 20 Supp ose that V i , i = 1 , . . . , k ar e close d semi m o dules in R n max , m with zer o interse ction. Then ther e e x i st close d ar ch i m e de an semimo dules V ′ i , i = 1 , . . . , k with zer o interse ction and such that e ac h V ′ i c ontains V i . Pr o of . In ev ery semimo dule V i , w e find a v ector y i with maximal supp o rt and suc h that || y i || = max( y i 1 , . . . , y i n ) = 1. F or all scalars δ > 0, define z i ( δ ) = y i ⊕ δ M j / ∈ supp( y i ) e j (42) and the semimo dules V i ( δ ) = { x | x = v ⊕ λz i , v ∈ V ′ i , λ ∈ R + } . (4 3) 15 These semimo dules are closed, as all arit hmetical op erations ar e contin uous. W e sho w tha t for δ > 0 small enough these semimo dules hav e zero in tersection. Assume by con tradiction that fo r all δ > 0, there exists a nonzero v ector u ( δ ) in the inters ection V 1 ( δ ) ∩ · · · ∩ V k ( δ ). After normalizing u ( δ ) , w e ma y assume that || u ( δ ) || = 1. F or a ny i = 1 , . . . , k and any δ w e ha v e tha t u ( δ ) = v i ( δ ) ⊕ λ i ( δ ) y i ⊕ λ i ( δ ) δ M j / ∈ supp( y i ) e j , (44) where v i ( δ ) is a ve ctor fro m V ′ i and λ i ( δ ) is a scalar. As | | u ( δ ) || = 1 and || y i || = 1, w e ha v e that λ i ( δ ) ≤ 1. So, there exists a sequence ( δ m ) m ≥ 1 con v erging t o 0 suc h that fo r all 1 ≤ i ≤ k , λ i ( δ m ) conv erges to a limit as m tends to infinity . Then w := lim m →∞ u ( δ m ) = lim m →∞ v i ( δ m ) ⊕ λ i ( δ m ) y i for all i . As V i are closed, w b elongs to V i at all i . Since | | w || = 1, w is not equal to 0 , whic h is a con tradiction. The follow ing is an immediate corolla ry of Theorem 19 and Prop osition 20. Theorem 21 (Separation theorem) If V i , i = 1 , . . . , k ar e cl o s e d s e mimo dules with zer o interse ction, then ther e exi s t a r chime de an halfsp ac es H i , i = 1 , . . . , k , which c ontain the c orr e s p onding semimo dules V i and ha ve zer o interse ction . The follo wing separation theorem for tw o closed semimo dules is a corolla r y of Theorem 21. Theorem 22 If U and V ar e two close d sem i m o dules wi th zer o interse ction, then ther e exists a close d halfsp ac e H U , which c ontains U and has zer o interse ction with V , and ther e exists a clo se d halfsp ac e H V , which c ontains V and h as zer o interse ction with U . As a consequence of Theorem 21, w e further deduc e a separation theorem for con v ex subsets of R n max , m . W e recall here some definitions from idemp o ten t conv ex g eometry , see e.g. [12] A subset C ⊂ R n max , m is c o nvex if λu ⊕ µv ∈ C , for all u, v ∈ C and λ, µ ∈ R max , m suc h that λ ⊕ µ = 1. The r e c ess i o n c one of a con ve x set C , rec( C ), is the set o f v ectors u suc h that v ⊕ λu ∈ C for all λ ∈ R max , m , where v is an arbitrary v ector of C . As sho wn in Prop. 2.6 of [12], if C is closed, the recession cone is independen t of the c ho ice of v . Observ e that when C is compact, its recession cone is zero. A set H aff giv en by H aff = { x | u /x ∧ α ≥ v /x ∧ γ } (45) with u , v ∈ R n max , m , u ≤ v , α , γ ∈ R max , m , α ≤ γ , will b e called (ide m p otent) affine halfsp a c e . It is called ar chime de an , if u , v , α and γ are p ositiv e. 16 F or a con vex set C ⊂ R n max , m define V ( C ) ⊂ R n +1 max ,m to b e the semimodule of v ectors of the form ( x 1 λ, . . . , x n λ, λ ) with x = ( x 1 , . . . , x n ) ∈ C and λ ∈ R max , m . Theorem 23 (Separation of con v ex sets) L et C 1 , . . . , C k b e close d c onvex subsets of R n max ,m with empty interse ction, and assume that the interse ction of the r e c ession c ones of C 1 , . . . , C k is zer o. Then, ther e exist affin e ar chime de an ha lfsp ac es H aff 1 , . . . , H aff k which c on tain the c orr e s p onding c onvex sets C i , i = 1 , . . . , k and have empty interse ction. Pr o of . F rom Prop. 2.1 6 of [12], w e know that the closure o f V ( C i ), V ( C i ), is equal to V ( C i ) ∪ (rec( C i ) × { 0 } ). Hence, the assumptions imply that the in tersection of V ( C 1 ) , . . . , V ( C k ) is zero. By Theorem 21, we can find archim edean halfspaces H i ⊃ V ( C i ) with zero in tersection. Ev ery H i can b e written as H i = { ( x 1 , . . . , x n , µ ) | u i /x ∧ α i /µ ≥ v i /x ∧ γ i /µ } ∪ { 0 } (46) with u i ≤ v i and α i ≤ γ i , understanding that x := ( x 1 , . . . , x n ). Observ e tha t for all x ∈ C i , ( x, 1) ∈ V ( C i ) ⊂ H i . W e deduce that the affine arc himedean halfspace H aff i = { x | u i /x ∧ α i ≥ v i /x ∧ γ i } (47) con tains C i . Since the in t ersection of t he half spaces H i is zero, t he intersec tion of the affine halfspaces H aff i m ust b e empty . In con v ex analysis, o ne can find an analogous separation theorem for sev eral compact con v ex sets, see [11], pages 39-40. W e no w deduce an idemp oten t analogue of the classical Helly’s Theorem. As observ ed b y S. Gaub ert and F. Meunier [13], t here is another pro of of this theorem, whic h is based on the direct idemp oten t a na lo gue of Radon’s argumen t (see [1 1]). Theorem 24 (Helly’s Theorem) Supp ose that V i , i = 1 , . . . , m is a c o l le ction of m ≥ n semimo d ules in R n max , m . If e ach n semim o dules interse ct nontrivial ly, then the whole c ol le c- tion has a n ontrivial in terse ction. Pr o of . It suffices to consider the case where the semimo dules V i are all closed. Indeed, the assumption implies that for all j := ( j 1 , . . . , j n ) ∈ { 1 , . . . , m } n , w e can choose a non-zero elemen t z j in the in tersection V j 1 ∩ · · · ∩ V j n . Let V ′ i denote the semimo dule generated b y the elemen ts z j that b elong to V i . The collection of semimo dules V ′ i , i = 1 , . . . , m still has the prop ert y that eac h n semimo dules in tersect nontrivially . Moreov er, V ′ i is closed, b ecause it is finitely generated (see e.g. Lemma 2 .20 of [12] or Coro llary 27 of [4]). Hence, if the 17 conclusion of t he theorem holds fo r closed semimo dules, we deduce that the whole collection V ′ i , i = 1 , . . . , m has a nontrivial in tersection, and since V i ⊃ V ′ i , the conclusion of the theorem also holds without an y closure assumption. In the discus sions that follo w, the semimo dules V i are all closed. W e argue b y contradiction, assuming that the whole collection has zero in tersection. By Theorem 20, w e can also assume that the semimo dules V i are arc himedean. F o r some n umber k < m ev ery k semimo dules in tersect nontrivially , but there are k + 1 semimodules, sa y V 1 , . . . , V k +1 , whic h ha v e zero in tersection. By Theorem 19, there exists a p ositiv e v ector y = y 0 and a scalar λ < 1 suc h that P k +1 · · · P 1 y ≤ λy . (48) F or all i we denote y i = P i · · · P 1 y 0 , where pro jectors are indexed mo dulo ( k + 1). By the homogeneit y and isotonicity of pro jectors, w e hav e that P l + k +1 · · · P l +1 y l ≤ λy l (49) for all l = 1 , . . . . Consider the v ectors z l = P l + k · · · P l +1 y l (50) for l = 1 , . . . , k + 1. Since eac h k semimo dules in tersect non trivially , the v ector z l m ust ha v e at least one co ordinate equal to that of y l , for otherwise y l w ould satisfy the first condition of Theorem 19, giving a contradiction. As k ≥ n , there are at least t wo n um b ers l and at least one n umber i such that z l has the same i th coo rdinate as y l . If w e tak e t he smallest of these t w o l num b ers, then ( P l + k +1 · · · P l +1 y l ) i = y l i . (51) But t his con tradicts (49). Hence an y k + 1 semimodules inte rsect nontrivially , whic h is a g ain a con tradiction. The theorem is pro v ed. There is also an affine v ersion o f this theorem. Theorem 25 Supp ose that C i , i = 1 , . . . , m is a c ol le ction of m ≥ n + 1 c onvex subsets of R n max ,m . If e ach n + 1 of these c onvex sets have a no nempty interse ction, then the whole c ol le ction h as a no n empty interse ction. Pr o of . Consider the semimodules V ( C 1 ) , . . . , V ( C m ) defined a b ov e, and apply Theorem 24 to them. 18 5 Ac kno wledge men ts The t w o a uthors thank Pe ter Butk o vi ˇ c and Hans Sc hneider for illuminating discussions whic h ha v e b een at the origin of the presen t work. The first author a lso thanks F r ´ ed ´ eric Meunier for ha ving drawn his a tten tion to t he max-plus analogues of Helly-t yp e t heorems. The second author is grateful to Andre ˘ ı Sob olevski ˘ ı for v aluable ideas a nd discuss ions concerning the analogy b etw een conv ex geometry and idemp oten t analysis. References [1] F.L. Baccelli, G . Cohen, G.J. Olsder, and J.P . Quadrat. Synch r onization and Line arity . Wiley , Chic hester, New Y ork, 1 992. [2] G. Birkhoff. L attic e the ory . Pro vidence, RI, 196 7. [3] H.H. Bausc hke , J.M. Bor wein, and A.S. Lewis. 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