Distance functions on convex bodies and symplectic toric manifolds
DDISTANCE FUNCTIONS ON CONVEX BODIES ANDSYMPLECTIC TORIC MANIFOLDS
H. FUJITA, Y. KITABEPPU, A. MITSUISHI
Abstract.
In this paper we discuss three distance functions on the set of con-vex bodies. In particular we study the convergence of Delzant polytopes, whichare fundamental objects in symplectic toric geometry. By using these obser-vations, we derive some convergence theorems for symplectic toric manifoldswith respect to the Gromov-Hausdorff distance.
Contents
1. Introduction 12. Three distance functions on the set of convex bodies 32.1. L -Wasserstein distance 32.2. Lebesgue volume 42.3. Hausdorff distance 43. Relation of distance functions 43.1. Equivalence among d W , d V and d H L -Wassersteindistance 16A.1. Weak convergence and Prokhorov’s theorem 17A.2. L -Wasserstein distance of probability measures 17Appendix B. Disintegration theorem 18References 181. Introduction
Convex polytopes, or more generally convex bodies, are classical and impor-tant objects in geometry. There are many results in which structures or proper-ties of convex polytopes are shown to have deep connections with other objects,through algebraic or combinatorial procedures. Among other such results, there isthe
Delzant construction [4], which is well known in symplectic geometry. Usingthe Delzant construction one obtains a natural bijective correspondence betweenthe set of
Delzant polytopes and the set of symplectic toric manifolds . Under thiscorrespondence, the geometric data of symplectic toric manifolds are encoded as
Mathematics Subject Classification.
Primary 53C23, Secondary 53D20, 52B12. a r X i v : . [ m a t h . M G ] M a r H. FUJITA, Y. KITABEPPU, A. MITSUISHI combinatorial or topological properties of their corresponding polytopes. For exam-ple, the cohomology ring of symplectic toric manifolds can be recovered completelyas the
Stanley-Reisner ring of the associated polytope. See e.g. [3] for more detailson this dictionary between Delzant polytopes and symplectic toric manifolds.The purpose of our project is to further develop aspects of this kind of cor-respondence from the viewpoint of Riemannian or metric geometry. The presentpaper contains two parts. Firstly, we establish relationships between three naturaldistance functions on the set of convex bodies. The first function d W is definedby the Wasserstein distance of probability measures associated with convex bod-ies. The Wasserstein distance is a quite important tool in recent developments ofgeometric analysis for metric measure spaces. The second distance d V is defined bythe Lebesgue volume of the symmetric difference of convex bodies. This distancefunction is natural from the viewpoint of symplectic geometry and is studied in [14]and [6]. The third function d H is the Hausdorff distance, which is a classical andbasic tool in geometry of convex bodies. The main result of the first part of thispaper is as follows. Theorem 1 (Theorem 3.1.3).
The metric topologies determined by the distancefunctions d W , d V and d H coincide with each other. Secondly, we investigate the relationship between the metric geometry of Delzantpolytopes and the Riemannian geometry of symplectic toric manifolds through theDelzant construction. Here we equip each symplectic toric manifold with a K¨ahlermetric called the
Guillemin metric [9], and we regard a symplectic toric manifoldas a Riemannian manifold. The main results in the second part of this paper arethe following.
Theorem 2 (Theorem 5.2.2).
For a sequence of Delzant polytopes { P i } i in R n , suppose that { P i } i converges to a Delzant polytope P in R n in the d H -topology(hence also in the d W -topology and d V -topology), and the limit of the numbers offacets of { P i } i coincides with that of P . Then the sequence of symplectic toric man-ifolds { M P i } i with the Guillemin metric converges to M P in the torus-equivariantGromov-Hausdorff topology. As a corollary (Corollary 5.2.3), we also have a torus-equivariant stability theo-rem in the setting of converging symplectic toric manifolds.
Theorem 3 (Theorem 5.3.1, Theorem 5.3.2).
For a sequence of Delzantpolytopes { P i } i in R n and a Delzant polytope P in R n , suppose that the corre-sponding sequence of symplectic toric manifolds { M P i } i converges to M P in thetorus-equivariant measured Gromov-Hausdorff topology. Then we have : • the fixed point set of M P i converges to that of M P . In particular we havethe lower semi-continuity of the Euler characteristic, and • we have a sequence which converges to P in d H -topology by using { P i } i andthe approximation maps for { M P i } i . We emphasize that there are no hypotheses on the curvature in the statementof the above theorem. By incorporating “potential functions”as in [1] we maytreat more general torus-invariant Riemannian metrics of symplectic toric manifoldswhich are not necessarily Guillemin metrics.In the present paper, we only consider the non-collapsing case. It is surelyinteresting to attack the same problems under collapsing limit, and we will discussthis in a subsequent paper. In addition, our general setting of convex bodies inthe first part of this paper is motivated by the fact that non-Delzant polytopes areincreasingly important in the context of toric degenerations of integrable systemsor projective varieties as in [10], [13] and so on.
ISTANCE FUNCTIONS ON CONVEX BODIES AND SYMPLECTIC TORIC MANIFOLDS 3
This paper is organized as follows. In Section 2 we introduce three distance func-tions on the set of convex bodies. In Section 3 we show that the three correspondingmetric topologies coincide. Note that the equivalence between the distance func-tion defined by the volume and the Hausdorff distance is classically known, by [15]for example. In [14] Pelayo-Pires-Ratiu-Sabatini studied several properties of themoduli space of Delzant polytopes with respect to the natural action of integralaffine transformations. This moduli space arises naturally from an equivalence re-lation of symplectic toric manifolds known as weak equivalence, and we expect itto be an important object in a subsequent research. We also give comments onthe distance function and the associated topology on this moduli space which werestudied in [6]. In Section 4 we discuss the definition of Delzant polytopes and thedescription of Guillemin metric on the corresponding symplectic toric manifolds. InSection 5 we discuss the relation between the convergence of Delzant polytopes andthe convergence of symplectic toric manifolds. In Appendix A we record severalfacts on probability measures and Wasserstein distance. In Appendix B we providea disintegration theorem which is important in the proof of Theorem 5.3.2.
Acknowledgement.
This work was partially done while the first author was vis-iting the Department of Mathematics, University of Toronto, and the Departmentof Mathematics and Statistics, McMaster University. He would like to thank bothinstitutions for their hospitality, especially for M. Harada. He is also grateful toY. Karshon and X. Tang for fruitful discussions. The first author is partly sup-ported by Grant-in-Aid for Scientific Research (C) 18K03288. The second author ispartly supported by Grant-in-Aid for Early-Career Scientists 18K13412. The thirdauthor is partly supported by Grant-in-Aid for Young Scientists (B) 15K17529and Scientific Research (A) 17H01091. Finally, the authors would like to expressgratitude to K. Ohashi who gave us a chance to begin this research.
Notations.
For a metric space (
X, d ), a subset Y of X , a point x in X and apositive real number r we use the following notations. • B ( x, r ) := { y ∈ X | d ( x, y ) < r } : open ball of radius r centered at x . • B ( Y, r ) := (cid:26) y ∈ X (cid:12)(cid:12)(cid:12)(cid:12) inf y (cid:48) ∈ Y d ( y, y (cid:48) ) < r (cid:27) : open r -neighborhood of Y . • dist( x, A ) := inf { d ( x, y ) | y ∈ A } : distance between x and A . • Diam( A ) := sup { d ( y, y (cid:48) ) | y, y (cid:48) ∈ A } : diameter of A .We use the notation (cid:107) · (cid:107) (resp. (cid:104)· , ·(cid:105) ) for the Euclidean norm (resp. inner product)on the Euclidean spaces. We also use the notation | A | for the Lebesgue measure ofa Lebesgue measurable subset A .2. Three distance functions on the set of convex bodies
Let C n be the set of all convex bodies in R n , i.e., C n is the set of all boundedclosed convex sets obtained as closures of open subsets in R n .2.1. L -Wasserstein distance. For each C ∈ C n let m C be the probability mea-sure on R n with compact support defined by m C := χ C H n ( C ) H n , where χ C is the characteristic function of C and H n is the n -dimensional Hausdorffmeasure on R n . Of course H n is equal to the n -dimensional Lebesgue measure L n ,however, since we put on the field of view of collapsing phenomena of convex bodiesinto lower dimensional objects, we prefer to use the Hausdorff measure. H. FUJITA, Y. KITABEPPU, A. MITSUISHI
Definition 2.1.1.
Define a function d W : C n × C n → R ≥ by d W ( C , C ) := W ( m C , m C ) , where W is the L -Wasserstein distance on the set of all probability measures on R n with finite quadratic moment.See Appendix A for basic definitions and facts on L -Wasserstein distance. Lemma 2.1.2. d W is a distance function on C n .Proof. Symmetricity, triangle inequality and non-negativity are clear. The non-degeneracy follows from the equivalence between the conditions d W ( C , C ) = W ( m C , m C ) = 0 and C = supp ( m C ) = supp ( m C ) = C . (cid:3) Lebesgue volume.
For C , C ∈ C n , let d V ( C , C ) be the Lebesgue volumeof the symmetric difference C (cid:52) C := ( C \ C ) ∪ ( C \ C ) : d V ( C , C ) := | C (cid:52) C | = (cid:90) R n χ C (cid:52) C ( x ) L n ( dx ) . This d V is indeed a distance function on C n and used in a study of convex bodiesclassically. See [5] or [15] for example.2.3. Hausdorff distance.
Let d H be the Hausdorff distance on the set of all com-pact subsets in R n .We also denote the restriction of d H to C n by the same letter d H : d H ( C , C ) := max { max x ∈ C min y ∈ C (cid:107) x − y (cid:107) , max y ∈ C min x ∈ C (cid:107) x − y (cid:107)} ( C , C ∈ C n ) . Relation of distance functions
Equivalence among d W , d V and d H . It is known that two distance func-tions d V and d H give the same metric topology. More precisely in [15] it is shownthat a sequence { P i } i in C n converges to Q ∈ C n in d V if and only if it convergesto Q in d H . Lemma 3.1.1.
For a sequence { P i } i in C n and Q ∈ C n , if d V ( P i , Q ) → i → ∞ ) ,then we have d W ( P i , Q ) → i → ∞ ) .Proof. Since lim i →∞ d V ( P i , Q ) = 0 implies lim i →∞ d H ( P i , Q ) = 0 we may assume that K i := Diam( P i ) ≤ K := 100 Diam( Q ) , and | log( | P i | / | Q | ) | < (cid:15) for small (cid:15) > i large enough. Now we definecouplings ξ i ∈ Cpl ( m P i , m Q ) ( i = 1 ,
2) by ξ ( X × X ) := m Q ( X ∩ X ∩ P i ∩ Q ) m P i ( X ) + m Q ( X \ ( X ∩ P i )) m P i ( X )when | Q | ≥ | P i | and ξ ( X × X ) := m Q ( X ) m P i ( X ∩ X ∩ P i ∩ Q ) + m Q ( X ) m P i ( X \ ( Q ∩ X ))when | P i | ≥ | Q | . Then we have ISTANCE FUNCTIONS ON CONVEX BODIES AND SYMPLECTIC TORIC MANIFOLDS 5 d W ( P i , Q ) ≤ (cid:115)(cid:90) R n × R n (cid:107) x − y (cid:107) ξ ( dx, dy ) + (cid:115)(cid:90) R n × R n (cid:107) x − y (cid:107) ξ ( dx, dy ) ≤ (cid:115) | Q \ P i || Q | · (101 K ) + (cid:115) | P i \ Q || P i | · (101 K ) ≤ · K (cid:115) | Q (cid:52) P i | min {| Q | , | P i |}≤ · K (cid:115) d V ( Q, P i ) e − (cid:15) | Q | → i → ∞ ) . (cid:3) Lemma 3.1.2.
For a sequence { P i } i in C n and Q ∈ C n , if d W ( P i , Q ) → i → ∞ ) ,then we have d V ( P i , Q ) → i → ∞ ) .Proof. Suppose that d W ( P i , Q ) → i → ∞ ). Then, m i := m P i converges weaklyto m := m Q , in particular, we have m i ( Q ) = | P i ∩ Q || P i | → m ( Q ) = 1by Theorem A.1.2. Since | P i ∩ Q | ≤ | Q | we have | P i | is bounded, and hence, | P i || Q | < c for some c >
0. Corollary A.2.3 implies that for two probability measures m i and m there exist a sequence of Borel measurable maps { T i : R n → R n } i such that( id × T i ) ∗ m ∈ Opt ( m, m i ) for all i and m ( { x ∈ Q | (cid:107) x − T i ( x ) (cid:107) ≥ a } ) = m ( { x ∈ R n | (cid:107) x − T i ( x ) (cid:107) ≥ a } ) → i → ∞ )for all a >
0. Let us fix an arbitrary positive number (cid:15) and set ξ := (cid:15) ( c + 1)( | Q | + 1) . Choose η small enough so that | B ( Q, η ) \ Q | < ξ. There exists N ∈ N such that m ( { x ∈ Q | (cid:107) T i ( x ) − x (cid:107) ≥ η } ) < ξ for all i ≥ N . Take and fix i > N . For x ∈ Q we put r ix := (cid:107) x − T i ( x ) (cid:107) . Then wehave Q ⊂ (cid:91) x ∈ Q B ( x, r ix ). We put U i := (cid:91) x ∈ Q,r ix ≤ η B ( x, r ix ) . We have | U i \ Q | ≤ | B ( Q, η ) \ Q | < ξ, | Q \ U i | = | Q | m ( Q \ U i ) ≤ | Q | m ( { x ∈ Q | (cid:107) x − T i ( x ) (cid:107) ) ≥ η } ) < | Q | ξ, H. FUJITA, Y. KITABEPPU, A. MITSUISHI and hence, | Q (cid:52) U i | < ( | Q | + 1) ξ . On the other hand we have | P i \ U i | = | P i | m i ( P i \ U i )= | P i | ( T i ) ∗ m ( P i \ U i )= | P i | m ( T − i ( P i ) \ T − i ( U i )) . Since ( T i ) ∗ m = m i we have that T − i ( P i ) = Q ( m -a.e.). This fact and T − i ( B ( x, r ix )) (cid:51) x imply that T − i ( U i ) ⊃ { x ∈ Q | (cid:107) x − T i ( x ) (cid:107) ≤ η } . In particular we have | P i \ U i | ≤ | P i | m ( { x ∈ Q | (cid:107) x − T i ( x ) (cid:107) > η } ) ≤ | P i | ξ. Similarly we have | U i \ P i | = | P i | m i ( U i \ P i ) = | P i | m ( T − i ( U i ) \ Q ) ≤ | P i | m ( B ( Q, η ) \ Q ) = | P i || Q | | B ( Q, η ) \ Q | < | P i || Q | ξ ≤ cξ, and hence | U i (cid:52) P i | ≤ ( | P i | + c ) ξ . Therefore we have d V ( P i , Q ) = | Q (cid:52) P i | ≤ | Q (cid:52) U i | + | U i (cid:52) P i |≤ ( | Q | + | P i | + c + 1) ξ ≤ ((1 + c ) | Q | + c + 1) ξ = (1 + c )( | Q | + 1) ξ = (cid:15). Since (cid:15) > d V ( P i , Q ) → (cid:3) As a corollary of Lemma 3.1.1 and Lemma 3.1.2 we have the following by Kra-towski’s axiom and the coincidence between the metric topology of d V and d H asshown in [15]. Theorem 3.1.3.
Three metric topologies on C n determined by d W , d V and d H coincide with each other. Moduli space of convex bodies and its topology.
We introduce themoduli space of convex bodies following [6] and [14]. Let G n := AGL( n, Z ) be theintegral affine transformation group. Namely G n is the direct product GL( n, Z ) × R n as a set and the multiplication on G n is defined by( A , t ) · ( A , t ) := ( A A , A t + t )for each ( A , t ) , ( A , t ) ∈ G n . This group G n acts on C n in a natural way, and C ∈ C n and C (cid:48) ∈ C n are called G n -congruent if C and C (cid:48) are contained in the same G n -orbit. Definition 3.2.1.
The moduli space of convex bodies (cid:101) C n with respect to the G n -congruence is defined by the quotient (cid:101) C n := C n /G n . Let π be the natural projection from C n to (cid:101) C n . Definition 3.2.2.
Define a function D V : (cid:101) C n × (cid:101) C n → R by D V ( α, β ) := inf { d V ( P , P ) | π ( P ) = α, π ( P ) = β } for ( α, β ) ∈ (cid:101) C n × (cid:101) C n . Theorem 3.2.3 ([6]) . D V is a distance function on (cid:101) C n and its metric topologycoincides with the quotient topology induced from π . ISTANCE FUNCTIONS ON CONVEX BODIES AND SYMPLECTIC TORIC MANIFOLDS 7
This G n -action and the moduli space (cid:101) C n arise naturally in the context of thegeometry of symplectic toric manifolds. In the subsequent sections we will discussfrom such point of view. Remark 3.2.4.
As it is noted in [6] we can not define a distance function on (cid:101) C n byusing the infimum of d H (or d W ) among all representatives, though, one may hopethat by considering infimum of d H among only “standard”representatives we can de-fine a distance function on (cid:101) C n . One possible candidates of “standard”representativesare the minimum variance (or quadratic moment) elements in the following sense.For each C ∈ C n define its variance by Var ( C ) := 1 | C | (cid:90) C (cid:107) x − b ( C ) (cid:107) L n ( dx ) , where b ( C ) is the barycenter of C which is determined uniquely by the condition (cid:104) b ( C ) , y (cid:105) = (cid:90) R n (cid:104) x, y (cid:105)L n ( dx )for any y ∈ R n . See [16] for example. The minimum variance element C ∈ C n is anelement of argmin { Var ( C (cid:48) ) | C (cid:48) ∈ C n is G n -congruent to C } . One can see that for any C ∈ C n there exist at least one and finitely many minimumvariance elements which have the common barycenter are G n -congruent to C .4. Delzant polytopes and symplectic toric manifolds
Delzant polytopes, symplectic toric manifolds and their moduli space.Definition 4.1.1.
A convex polytope P in R n is called a Delzant polytope if P satisfies the following conditions : • P is simple, that is, each vertex of P has exactly n edges. • P is rational, that is, at each vertex all directional vectors of edges can betaken as integral vectors in Z n . • P is smooth, that is, at each vertex we can take integral directional vectorsof edges as a Z -basis of Z n in R n .We denote the subset of C n consisting of all Delzant polytopes by D n and definetheir moduli space by (cid:101) D n := D n /G n .Recall that a symplectic toric manifold ( M, ω, ρ, µ ) is a data consisting of • a compact connected symplectic manifold ( M, ω ) of dimension 2 n , • a homomorphism ρ from the n -dimensional torus T n to the group of sym-plectomorphisms of M which gives a Hamiltonian action of T n on M and • a moment map µ : M → R n = (Lie( T n )) ∗ .The famous Delzant construction gives a correspondence between Delzant poly-topes and symplectic toric manifolds. Theorem 4.1.2 ([12]) . The Delzant construction gives a bijective correspondencebetween (cid:101) D n and the set of all weak isomorphism classes of n -dimensional symplec-tic toric manifolds. Here two symplectic toric manifolds ( M , ω , ρ , µ ) and ( M , ω , ρ , µ ) are weakly isomorphic if there exist a diffeomorphism f : M → M and a groupisomorphism φ : T n → T n such that f ∗ ω = ω and ρ ( g )( x ) = ρ ( φ ( g ))( f ( x )) for all ( g, x ) ∈ T n × M . In [12] the equivalence relation “weakly isomorphism ” is called just “equivalent ”. In thispaper we follow the terminology in [14].
H. FUJITA, Y. KITABEPPU, A. MITSUISHI
Based on the above fact the moduli space (cid:101) D n is also called the moduli space oftoric manifolds in [14]. In [14] they show that ( D n , d V ) is neither complete norlocally compact and (cid:101) D is path connected.4.2. Brief review on the Delzant construction.
For later convenience we givea brief review on the Delzant construction here.Let P be an n -dimensional Delzant polytope and(4.2.1) l ( r ) ( x ) := (cid:104) x, ν ( r ) (cid:105) − λ ( r ) = 0 ( r = 1 , · · · , N )a system of defining affine equations on R n of facets of P , each ν ( r ) being inwardpointing normal vector of r -th facet and N is the number of facets of P . In otherwords P can be described as P = N (cid:92) r =1 { x ∈ R n | l ( r ) ( x ) ≥ } . We may assume that each ν ( r ) is primitive and they form a Z -basis of Z n . Considerthe standard Hamiltonian action of the N -dimensional torus T N on C N with themoment map˜ µ : C N → ( R N ) ∗ = Lie( T N ) ∗ , ( z , . . . , z N ) (cid:55)→ − ( | z | , . . . , | z N | ) + ( λ (1) , . . . , λ ( N ) ) . Let ˜ π : R N → R n be the linear map defined by e r (cid:55)→ ν ( r ) , where e r ( r = 1 , . . . , N )is the r -th standard basis of R N . Note that ˜ π induces a surjection ˜ π : Z N → Z n between the standard lattices by the last condition in Definition 4.1.1, and henceit induces surjective homomorphism between tori, still denoted by ˜ π ,˜ π : T N = R N / Z N → T n = R n / Z n . Let H be the kernel of ˜ π which is an ( N − n )-dimensional subtorus of T N and h itsLie algebra. We have exact sequences1 → H ι → T N ˜ π → T n → , → h ι → R N ˜ π → R n → → ( R n ) ∗ ˜ π ∗ → ( R N ) ∗ ι ∗ → h ∗ → , where ι is the inclusion map. Then the composition ι ∗ ◦ ˜ µ : C N → h ∗ is theassociated moment map of the action of H on C N . It is known that ( ι ∗ ◦ ˜ µ ) − (0)is a compact submanifold of C N and H acts freely on it. We obtain the desiredsymplectic manifold M P := ( ι ∗ ◦ ˜ µ ) − (0) /H equipped with a natural Hamiltonian T N /H = T n -action. Note that the standard flat K¨ahler structure on C N induces aK¨ahler structure on M P . The associated Riemannian metric is called the Guilleminmetric . Remark 4.2.1.
In the above set-up we assume that the number of facets of P , say N , is equal to that of the defining inequalities, though, it is possible to considerthe similar construction formally for any system of inequalities which has morethan N inequalities. Such a construction may produce a symplectic toric manifoldequipped with metric which is not isometric to the Guillemin metric.There exists an explicit description of the Guillemin metric. We give the de-scription following [1]. Consider a smooth function(4.2.2) g P := 12 N (cid:88) r =1 l ( r ) log l ( r ) : P ◦ → R , An integral vector u in R n is called primitive if u cannot be described as u = ku (cid:48) for anotherintegral vector u (cid:48) and k ∈ Z with | k | > ISTANCE FUNCTIONS ON CONVEX BODIES AND SYMPLECTIC TORIC MANIFOLDS 9 where P ◦ is the interior of P . It is known that M ◦ P := µ − P ( P ◦ ) is an open densesubset of M P on which T n acts freely and there exists a diffeomorphism M ◦ P ∼ = P ◦ × T n . Under this identification ω P | M ◦ P can be described as ω P | M ◦ P = dx ∧ dy = n (cid:88) i =1 dx i ∧ dy i using the standard coordinate ( x, y ) = ( x , . . . , x n , y , . . . , y n ) ∈ P ◦ × T n . Thecoordinate on M ◦ P induced from ( x, y ) ∈ P ◦ × T n is called the symplectic coordinate on M P . Theorem 4.2.2 ([9]) . Under the symplectic coordinates ( x, y ) ∈ P ◦ × T n ∼ = M ◦ P ⊂ M P , the Guillemin metric can be described as (cid:18) G P G − P (cid:19) , where G P := Hess x ( g P ) = (cid:18) ∂ g P ∂x k ∂x l (cid:19) k,l =1 ,...,n is the Hessian of g P . Remark 4.2.3. If P and P (cid:48) in D n are G n -congruent, then the correspondingRiemannian manifolds M P and M P (cid:48) are isometric to each other. In fact as it isnoted in [1, Section 3.3], for ϕ ∈ G n we have an isomorphism between M P and M ϕ ( P ) as K¨ahler manifolds. The isomorphism is induced by the map P × T → ϕ ( P ) × T , ( x, t ) (cid:55)→ ( ϕ ( x ) , (( ϕ ∗ ) − ) T ( t )), where ( ) T is the transpose and ϕ ∗ is theautomorphism of T which is induced by ϕ .5. Convergence of polytopes and symplectic toric manifolds
Hereafter we do not often distinguish a sequence itself and a subsequence of it.5.1.
Convergence of polytopes and related quantities.
For a polytope P in R n we denote the set of all k -dimensional faces of P by { F ( r ) k ( P ) } r . In particularwe denote the set of all facets by { F ( r ) ( P ) } r . We often omit the superscript r forsimplicity and denote each face by F k ( P ) for example. Proposition 5.1.1.
For a sequence { P i } i ⊂ D n suppose that d H ( P i , P ) → i →∞ ) for P ∈ D n . Then for any x ∈ F ( P ) there exists a sequence { x i ∈ F ( P i ) } i suchthat x i → x ( i → ∞ ) .Proof. For x ∈ F ( P ) suppose thatlim sup i →∞ dist( x, ∂P i ) > (cid:15) for some (cid:15) >
0. We may assume that B ( x, (cid:15) ) ∩ ∂P i = ∅ for any i by taking a subsequence. By the above assumption and P i → P in d H there exists a sequence { y i ∈ P ◦ i } i such that y i → x . For any i large enough wemay assume that (cid:107) x − y i (cid:107) < (cid:15) . Then we havedist( y i , ∂P i ) ≥ dist( x, ∂P i ) − (cid:107) x − y i (cid:107) ≥ (cid:15), and hence, B ( y i , (cid:15) ) ∩ ∂P i = ∅ . In particular we have B ( y i , (cid:15) ) ⊂ P ◦ i . Let ν be an inward unit normal vector of F ( P ) and put z i := y i − (cid:15)ν. Here we regard T = T n = ( S ) n and S = R / Z . We have z i ∈ B ( y i , (cid:15) ) ⊂ P ◦ i and it converges to z := x − (cid:15)ν . On the otherhand one can see that B ( z, (cid:15) ) ⊂ P c and z ∈ lim i →∞ P i = P . It contradicts to d H ( P i , P ) → i → ∞ ). (cid:3) Corollary 5.1.2.
As in the same setting in Proposition 5.1.1 for any k = 0 , , . . . , n − and a point x ∈ F k ( P ) there exists a sequence { x i ∈ F k ( P i ) } i such that x i → x ( i → ∞ ) .Proof. For any x ∈ F n − ( P ) let F ( P ) be a facet of P which contains x ∈ F n − ( P ).By Proposition 5.1.1 F ( P ) can be described as a limit of a union of facets of F ( P i ).The proof of 5.1.1 shows that F n − ( P ) can be described as a limit of ( n − F ( P i ). One can prove the claim in an inductive way. (cid:3) Corollary 5.1.3.
As in the same setting in Proposition 5.1.1 the number of k -dimensional faces is lower semi-continuous for any k : { F ( r ) k ( P ) } r ≤ lim i →∞ ( { F ( r ) k ( P i ) } r ) . Corollary 5.1.4.
Consider the same setting in Proposition 5.1.1. For any facet F ( r ) ( P ) , its normal vector ν ( r ) and a scalar λ ( r ) there exists a sequence of facet F ( r i ) ( P i ) such that the corresponding defining affine functions converges to that of F ( P ) , i.e., l ( r i ) i → l ( r ) ( i → ∞ ) .Proof. By Proposition 5.1.1, for any facet F ( r ) ( P ) of P , one can take a sequenceof facets { F ( r i ) ( P i ) } i of P i which converges to F ( r ) ( P ). We may assume that thesequence of unit normal vectors of F ( r i ) ( P i ) converges to that of F ( r ) ( P ). It impliesthat the corresponding defining affine functions l ( r i ) i converge to l ( r ) . (cid:3) We say a sequence of k -dimensional faces { F k ( P i ) } i of a sequence { P i } i in D n converges essentially to a k -dimensional face F k ( P ) of P ∈ D n iflim i →∞ H k ( F k ( P i )) > i →∞ d H ( F k ( P i ) , F ) = 0for a closed subset F of F k ( P ).Next we consider the 2-dimensional case D . Theorem 5.1.5.
For a sequence { P i } i ⊂ D suppose that d H ( P i , P ) → i → ∞ ) for some P ∈ D . For each facet F ( r ) ( P ) of P and its primitive normal vector ν ( r ) ,there exists a sequence of primitive normal vectors { ν ( r i ) i } i of F ( r i ) ( P i ) such that ν ( r i ) i → ν ( r ) ( i → ∞ ) .Proof. By Corollary 5.1.3 and the semi-continuity of the Hausdorff measure in thenon-collapsing limit we may assume that for each facet (=edge) F ( r ) ( P ) there existsa sequence { F ( r i ) ( P i ) } i of facets of { P i } i which converges essentially to F ( r ) ( P ).We rearrange the indices so that r = r i = 1 for all i . Moreover we may as-sume that the facets are numbered in a counterclockwise way. Note that by thesmoothness condition the determinant of the 2 × ± { F (1) ( P i ) } i converges essentially to F (1) ( P ) the sequence of inward unitnormal vectors converges : ν (1) i (cid:107) ν (1) i (cid:107) → ν (1) (cid:107) ν (1) (cid:107) ( i → ∞ ) . ISTANCE FUNCTIONS ON CONVEX BODIES AND SYMPLECTIC TORIC MANIFOLDS 11
Since { ν (1) i } i is a sequence of integral vectors it suffices to show that {(cid:107) ν (1) i (cid:107)} i is abounded sequence.Suppose that {(cid:107) ν (1) i (cid:107)} i is unbounded. In this case note that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) det (cid:32) ν (1) i (cid:107) ν (1) i (cid:107) , ν (2) i (cid:107) ν (2) i (cid:107) (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1 (cid:107) ν (1) i (cid:107)(cid:107) ν (2) i (cid:107) | det( ν (1) i , ν (2) i ) | ≤ (cid:107) ν (1) i (cid:107) → i → ∞ ) . It implies that the facets F (1) ( P i ) and F (2) ( P i ) tends to be parallel as i → ∞ . Thesame situation holds for any pair of adjacent facets of P i in which at least one ofthe sequence of primitive normal vectors is unbounded. Now since { P i } i convergesto a simple convex polytope P , there exist at least two facets { F ( r i ) ( P i ) } i and { F ( r (cid:48) i ) ( P i ) } i with r i ≤ r (cid:48) i such that they converge essentially to some facets of P which are adjacent to F (1) ( P ). If { ν ( r ) i } i are unbounded for r = 1 , , . . . , r i − { F ( r ) ( P i ) } i tend to beparallel to each other. It implies that F (1) ( P i ) and F ( r i ) ( P i ) tend to be parallel eachother, and it is a contradiction. It implies that there exists r ∈ { , . . . , r i − } suchthat ν ( r ) i is bounded. The same argument implies that there exists r (cid:48) ∈ { r (cid:48) i , . . . , N i } such that ν ( r (cid:48) i ) i is bounded, where N i is the number of facets of P i .Since { F ( r ) ( P i ) } i and { F ( r (cid:48) ) ( P i ) } i tend to be parallel to F (1) ( P ) and the boundedprimitive normal vectors { ν ( r ) i } i and { ν ( r (cid:48) ) i } i are integral vectors then we may as-sume that ν ( r i ) i = ν (1) i = ν ( r (cid:48) i ) i for any sufficiently large i (by taking a subsequence ofthe subsequence). It is a contradiction because such a situation cannot be realizedin a convex polytope P i . In particular {(cid:107) ν (1) i (cid:107)} i is bounded, and it completes theproof of the theorem. (cid:3) Remark 5.1.6.
In Theorem 5.1.5 the boundedness of each primitive normal vector { ν ( r ) i } i implies that it contains a constant subsequence.By the same argument we have the following convergence in the higher dimen-sional non-degenerate case. Theorem 5.1.7.
For a sequence { P i } i ⊂ D n suppose that d H ( P i , P ) → i → ∞ ) for some P ∈ D n and { F ( r ) ( P ) } r = lim i →∞ ( { F ( r ) ( P i ) } r ) . For each facet F ( r ) ( P ) of P and its primitive normal vector ν ( r ) , there exists a sequence of primitive normalvectors { ν ( r i ) i } i of F ( r i ) ( P i ) such that ν ( r i ) i → ν ( r ) ( i → ∞ ) .Proof. As in the proof of Theorem 5.1.5 we can take a sequence of primitive normalvectors { ν (1) i } i of { F (1) ( P i ) } i , and it suffices to show that {(cid:107) ν (1) i (cid:107)} i is bounded.Suppose that {(cid:107) ν (1) i (cid:107)} i is unbounded. Consider a vertex of F (1) ( P i ) and facetsaround it. We may assume that they are numbered as r = 2 , , · · · , n . Then fortheir primitive normal vectors we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) det (cid:32) ν (1) i (cid:107) ν (1) i (cid:107) , ν (2) i (cid:107) ν (2) i (cid:107) , · · · , ν ( n ) i (cid:107) ν ( n ) i (cid:107) (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:107) ν (1) i (cid:107) → i → ∞ ) . It contradicts to our assumption { F ( r ) ( P ) } r = lim i →∞ ( { F ( r ) ( P i ) } r ). (cid:3) From convergence of polytope to convergence of Guillemin metric.
We first give the definition of equivariant (measured) Gromov-Hausdorff conver-gence as a special case of [7, Definition 1-3].
Definition 5.2.1.
Let X = ( X, d ) be a compact metric space and { X i = ( X i , d i ) } i be a sequence of compact metric spaces. Suppose that there exists a group G whichacts on X and each X i in an effective and isometric way. Then { X i } i converges to X in the G - equivariant Gromov-Hausdorff topology if there exist sequences of maps { f i : X i → X } i , group automorphisms { ρ i : G → G } i and positive numbers { (cid:15) i } i such that the following conditions hold for any i large enough.(1) (cid:15) i → i → ∞ .(2) | d i ( x, y ) − d ( f i ( x ) , f i ( y )) | < (cid:15) i for all x, y ∈ X i .(3) For any p ∈ X there exists x ∈ X i such that d ( p, f i ( x )) < (cid:15) i .(4) d ( f i ( gx ) , ρ i ( g ) f i ( x )) < (cid:15) i for all x ∈ X i and g ∈ G .This situation will be denoted by X i G -eqGH −−−−−→ X (or X i → X for simplicity) and f i are called approximation maps .Moreover if X (resp. { X i } i ) is equipped with a G -invariant measure m (resp. m i ) in such a way that ( X, m ) (resp. ( X i , m i )) is a metric measure space andthe push forward measure ( f i ) ∗ m i converges to m weakly, then we say { ( X i , m i ) } i converges to ( X, m ) in the G -equivariant measured Gromov-Hausdorff topology andwe will denote X i G -eqmGH −−−−−−→ X .When X (resp. X i ) is a Riemannian manifold, we consider its Riemanniandistance.The above conditions (2), (3) and (4) mean that the approximation map f i is almost isometric, almost surjective and almost equivariant .As a corollary of Theorem 5.1.7 we have the following convergence theorem ofsymplectic toric manifolds. We emphasize that we do not put any assumptions oncurvatures in our theorem below. Theorem 5.2.2.
For a sequence { P i } i ⊂ D n suppose that d H ( P i , P ) → i → ∞ ) for P ∈ D n and { F ( r ) ( P ) } r = lim i →∞ ( { F ( r ) ( P i ) } r ) . Then there exists a subse-quence of { M P i } i which converges to M P in the T -equivariant Gromov-Hausdorfftopology.Proof. We use the same notations as in Section 4.2 with suffix i . We may as-sume N = { F ( r ) ( P ) } r = { F ( r ) ( P i ) } r = N i . The proof of Theorem 5.1.7implies that h i = h and H i = H for i (cid:29)
0. Moreover as a corollary of Theo-rem 5.1.7 we have λ ( r ) i → λ ( r ) ( i → ∞ ) for the constants of the defining equa-tions of P i (after renumbering the facets). As a consequence ( ι ∗ i ◦ ˜ µ i ) − (0) con-verges to ( ι ∗ ◦ ˜ µ ) − (0) in the equivariant Gromov-Hausdorff topology . Then { M P i = ( ι ∗ i ◦ ˜ µ i ) − (0) /H i } i converges to M P = ( ι ∗ ◦ ˜ µ ) − (0) /H in the Gromov-Hausdorff topology by [7, Theorem2-1]. Moreover the identifications H i = H induceidentifications T ni = T N /H i = T N /H = T n , which makes the above convergenceinto the T -equivariant Gromov-Hausdorff topology. (cid:3) Corollary 5.2.3.
Under the same assumptions in Theorem 5.2.2, take a subse-quence in { M P i } i which converges to M P . Then M P i are T -equivariantly diffeo-morphic to M P for i (cid:29) .Proof. By Theorem 5.1.5 we may assume that ν ( r ) i = ν ( r ) for i (cid:29)
0. On the otherhand each M P i is T -equivariantly diffeomorphic to the toric variety associated withthe fan Σ P i . Note that Σ P i is determined by the normal vectors { ν ( r ) i } r and it doesnot depend on { λ ( r ) i } r (See [3] for example). It implies the claim. (cid:3) Remark 5.2.4.
It can not be expected that a convergence as in Theorem 5.2.2occurs in general. Consider a sequence of Delzant pentagon { P i } i as in Figure 1,which converges to a rectangle P defined by 5 inequalities. It is known that the In fact this convergence is nothing other than the Hausdorff convergence of a sequence ofcompact subsets in R N . ISTANCE FUNCTIONS ON CONVEX BODIES AND SYMPLECTIC TORIC MANIFOLDS 13
Figure 1.
A sequence of pentagons which converges to a rectanglesymplectic toric manifolds correspond to each pentagon P i are (diffeomorphic to)a 1 point blow-up of C P × C P . The limiting process to P corresponds to shrinkthe exceptional divisor in M P i . Their limit as symplectic quotient is defined by5 inequalities, and it carries a Riemannian metric which is not isometric to theGuillemin metric. On the other hand in our setting M P is C P × C P equippedwith the Guillemin metric. To deal with these subtle phenomena we have to considerfiner structures on D n or (cid:101) D n and incorporate potential functions . We will discusssuch formulation in a subsequent paper.5.3. From convergence of Guillemin metrics to convergence of polytopes.
Now let us discuss the convergence of the opposite direction.Hereafter for each P ∈ D n we denote the symplectic toric manifold equippedwith the Guillemin metric by M P = ( M P , ω P ), and we use the Liouville volumeform vol M P := ( ω P ) ∧ n n ! on the symplectic toric manifold M P . In this way we think M P as a metric measure space. Theorem 5.3.1.
Let { P i } i be a sequence in D n . Suppose that a sequence ofsymplectic toric manifolds { M P i } i converges to M P for some P ∈ D n in the T -equivariant measured Gromov-Hausdorff topology. Let { f i : M P i → M } i be asequence of approximation maps of the convergence. If { P i } i are contained in asufficiently large ball in R n , then we have lim i →∞ f i ( M TP i ) = M TP , where M TP i and M TP are the fixed point sets of T -actions. In particular we have lim i →∞ χ ( M P i ) ≥ χ ( M P ) , where χ ( · ) denotes the Euler characteristic.Proof. For simplicity we denote M i := M P i and M := M P . We first show thatlim i →∞ f i ( x i ) ∈ M T for any sequence { x i ∈ M Ti } i . Suppose that there exists δ > f i ( x i ) / ∈ B ( M T , δ ) for infinitely many i . For (cid:15) >
0, we define δ (cid:15) as theminimal δ (cid:48) > y (cid:54)∈ B ( M T , δ (cid:48) ), then Diam( T · y ) ≥ (cid:15) . Note that since M is compact such δ (cid:15) > δ (cid:15) → (cid:15) →
0. Since f i is almost T -equivariantwe have (cid:15) i > d ( ρ i ( t ) f i ( x i ) , f i ( tx i )) = d ( ρ i ( t ) f i ( x i ) , f i ( x i )) for all t ∈ T , where { (cid:15) i } i is a sequence of positive numbers as in Definition 5.2.1 and d is the Riemannian distance of M . It implies that Diam( T · f i ( x i )) < (cid:15) i → i → ∞ . If we take i large enough so that δ (cid:15) i < δ , then we have f i ( x i ) ∈ B ( M T , δ (cid:15) i ).It contradicts to f i ( x i ) / ∈ B ( M T , δ ).Next we show that for any δ > i ∈ N such that f − i ( M T ) ⊂ B ( M Ti , δ )holds for all i > i . If not then there exists δ > x i ∈ f − i ( M T ) and x i / ∈ B ( M Ti , δ ) for infinitely many i . Since f i is almost isometry andalmost T -equivariant we have d i ( tx i , x i ) < d ( f i ( tx i ) , f i ( x i )) + (cid:15) i < d ( f i ( tx i ) , tf i ( x i )) + (cid:15) i < (cid:15) i for all t ∈ T , where d i is the Riemannian distance of M i . It implies Diam( T · x i ) < (cid:15) i . On the other hand it is known that each T · x i is a flat torus, and hence,Diam( T · x i ) → i → ∞ ) implies Vol( T · x i ) → i → ∞ ), where Vol is theRiemannian volume with respect to the induced Riemannian metric. Now con-sider a compact subset P (cid:48) i := µ i ( M i \ B ( M Ti , δ )) of P i . Since { M i } i converges to M in the measured Gromov-Hausdorff topology { Vol( M i ) } i converges to Vol( M ).Duistermaat-Heckman’s theorem implies that the Euclidean volumes of { P i } i con-verge to that of P . In particular they are bounded below by a positive constant.Moreover since we assume that { P i } i are contained in a ball, the sequence of convexpolytopes { P i } i converges to some convex body Q in the Hausdorff distance. Asin the same way { P (cid:48) i } i converges to some compact subset Q (cid:48) of Q . Let Q (0) be thelimit point set of µ i ( M Ti ) = P (0) i . Then we have Q (0) ∩ Q (cid:48) = ∅ . When we take δ (cid:48) > Q (0) , Q (cid:48) ) > δ (cid:48) we have dist( P (0) i , P (cid:48) i ) > δ (cid:48) . Theformula of volumes of the orbits in [11] implies thatlim inf i →∞ Vol( T · x i ) > . It contradicts to lim i →∞ Vol( T · x i ) = 0.The inequality lim i →∞ χ ( M P i ) ≥ χ ( M P ) , follows from the fact that the Euler characteristic of symplectic toric manifold isequal to the number of fixed points. (cid:3) Hereafter we discuss the convergence of polytopes under the same assumption inTheorem 5.3.1. We first take and fix a section S i : P i → M P i of the moment map µ i : M P i → P i for each i . Note that each S i is neither smooth nor continuous butonly measurable in general. Let { f i : M P i → M P } i be a sequence of approximationmaps. It is known that we may assume that f i is a Borel measurable map. Foreach i we define F i : P i → P by the composition F i := µ ◦ f i ◦ S i . Theorem 5.3.2.
Under the same assumptions in Theorem 5.3.1 there exists asubsequence of { F i ( P i ) } i which converges to P in d H topology. To show Theorem 5.3.2 we prepare two lemmas. Strictly speaking the formula in [11] can be applied when µ i ( x i ) is in the interior part of P i .So the above argument shows that { x i } i cannot be taken in such an interior part. As the nextstep we assume that { x i } i sits in the inverse image of the interior part of codimension one face,and we deduce the contradiction. We proceed the same step for higher codimension face. ISTANCE FUNCTIONS ON CONVEX BODIES AND SYMPLECTIC TORIC MANIFOLDS 15
Lemma 5.3.3.
Consider the same setting as in Theorem 5.3.2. For any ϕ ∈ C b ( R n ) there exists a sequence of measurable maps { ϕ i : P i → R } i such that lim i →∞ (cid:90) P i ϕ i d L n = (cid:90) P ϕ d L n . Proof.
Let µ i : M P i → P i ⊂ R n and µ : M → P ⊂ R n be the moment maps. ByDuistermaat-Heckman’s theorem we have ( µ i ) ∗ (vol M Pi ) = L n | P i .For ϕ ∈ C b ( R n ) we define ˜ ϕ ∈ C ( M P ) by ˜ ϕ := ϕ ◦ µ . Let { f i } i be a familyof approximation maps for M P i T -eqmGH −−−−−−→ M P . We define a sequence of mea-surable functions { ˜ ϕ i : M i → R } i by ˜ ϕ i := ˜ ϕ ◦ f i . Let { (vol M Pi ) y } y ∈ P i (resp. { (vol M P ) y } y ∈ P ) be a disintegration (See Appendix B) for µ i : M P i → P i (resp. µ : M P → P ) and define a sequence of measurable functions { ϕ i : P i → R } i by(5.3.1) ϕ i ( y ) := (cid:90) M Pi ˜ ϕ i ( x )(vol M Pi ) y ( dx ) . Since ( f i ) ∗ (vol M Pi ) converges to vol M P weakly we have (cid:90) P i ϕ i ( y ) L n ( dy ) = (cid:90) P i (cid:32)(cid:90) M Pi ˜ ϕ i ( x )(vol M Pi ) y ( dx ) (cid:33) L n ( dy )= (cid:90) M Pi ˜ ϕ i ( x )vol M Pi ( dx ) = (cid:90) M Pi ˜ ϕ ( f i ( x ))vol M Pi ( dx ) −−−→ i →∞ (cid:90) M P ˜ ϕ ( x )vol M P ( dx )= (cid:90) P (cid:18)(cid:90) M P ˜ ϕ ( x )(vol M P ) y ( dx ) (cid:19) L n ( dy )= (cid:90) P (cid:32)(cid:90) µ − ( y ) ϕ ( µ ( x ))(vol M P ) y ( dx ) (cid:33) L n ( dy )= (cid:90) P (cid:32)(cid:90) µ − ( y ) ϕ ( y )(vol M P ) y ( dx ) (cid:33) L n ( dy )= (cid:90) P ϕ ( y ) L n ( dy ) . (cid:3) Lemma 5.3.4.
As in the same setting in Theorem 5.3.2 we have lim i →∞ | P i | (cid:90) P i ϕ i d L n = lim i →∞ | P i | (cid:90) P i ϕ ◦ F i d L n . for any ϕ ∈ C b ( R n ) , where ϕ i are as in Lemma 5.3.3.Proof. Let { ρ i : T n → T n } i be a sequence of automorphisms as in Definition 5.2.1for M P i eq − m GH −−−−−−→ M P . Fix η > ϕ ∈ C b ( R n ). For any y ∈ P i we have(5.3.2) | ϕ i ( y ) − ϕ ( F i ( y )) | ≤ (cid:90) µ − i ( y ) | ϕ ( µ ( f i ( x ))) − ϕ ( µ ( f i ( S i ( y )))) | (vol M Pi ) y ( dx ) . Since for any x ∈ µ − i ( y ) there exists t x ∈ T such that x = t x · S i ( y ) we have (cid:107) µ ( f i ( x )) − µ ( f i ( S i ( y ))) (cid:107) = (cid:107) µ ( f i ( t x · S i ( y ))) − µ ( f i ( S i ( y ))) (cid:107) = (cid:107) µ ( f i ( t x · S i ( y ))) − µ ( ρ i ( t x ) · f i ( S i ( y ))) (cid:107) . On the other hand since ϕ and µ are uniformly continuous and { M P i } i convergesto M P in the T -equivariant Gromov-Hausdorff topology there exists i ∈ N suchthat if i > i , then | ϕ ( µ ( f i ( x ))) − ϕ ( µ ( f i ( S i ( y )))) | = | ϕ ( µ ( f i ( x ))) − ϕ ( µ ( ρ i ( t x ) · f i ( S i ( y )))) | < η. In particular we have | ϕ i ( y ) − ϕ ( F i ( y )) | < η in (5.3.2), and hence, 1 | P i | (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) P i ( ϕ i ( y ) − ϕ ( F i ( y ))) L n ( dy ) (cid:12)(cid:12)(cid:12)(cid:12) < η. Note that our assumption M P i T -eqmGH −−−−−−→ M P and Duistermaat-Heckman’s theoremimply | P i | = vol M Pi ( M P i ) → | P | = vol M P ( M P ). Since η > | P i | (cid:90) P i ϕ i ( y ) L n ( dy ) exists and we have the required equalitylim i →∞ | P i | (cid:90) P i ϕ i ( y ) L n ( dy ) = lim i →∞ | P i | (cid:90) P i ϕ ( F i ( y )) L n ( dy ) . (cid:3) Proof of Theorem 5.3.2.
Let ϕ ∈ C b ( R n ). By Lemma 5.3.3 and Lemma 5.3.4 wehave a sequence of measurable maps { F i : P i → P } i and measurable functions { ϕ i : P i → R } i such thatlim i →∞ (cid:90) P ϕ ( y )( F i ) ∗ ( L n )( dy ) = lim i →∞ (cid:90) P i ϕ i ( y ) L n ( dy ) = (cid:90) P ϕ ( y ) L n ( dy ) . Note that we have | P i | → | P | ( i → ∞ ) under our assumption, measured Gromov-Hausdorff convergence, and Duistermaat-Heckman’s theorem. This equality impliesthat the sequence of probability measures { ( F i ) ∗ m P i } i converges weakly to m P .Since F i ( P i ) ⊂ P we havelim R →∞ lim sup i →∞ (cid:90) R n \ B (0 ,R ) (cid:107) x (cid:107) ( F i ) ∗ m P i ( dx ) = 0 . It implies that W (( F i ) ∗ m P i , m P ) → i → ∞ ) by (1) and (2) in Theorem A.2.1,and hence, supp (( F i ) ∗ ( m P i )) = F i ( P i ) converges to P as i → ∞ . (cid:3) Remark 5.3.5.
Regarding Theorem 5.3.1 and Theorem 5.3.2 let us mention somecomments. It is natural to consider the following two problems; removing theassumption on uniformly boundedness of { P i } i and getting a convergence of { P i } i to P in the Gromov-Hausdorff or d H -topology. One can see that these are not truein the literal sense because of the ambiguity of the affine transformation groups G n . We could address these problems in terms of the moduli space. Namely onemay hope that if { M P i } i converges to M P in the T -equivariant measured Gromov-Hausdorff topology, then there exists a sequence { ϕ i } i in G n such that { ϕ i ( P i ) } i converges to P in the Gromov-Hausdorff or d H -topology. It would be useful toconsider minimum variance elements explained in Remark 3.2.4. Appendix A. Preliminaries on probability measures and L -Wasserstein distance In this appendix we summarize several facts on probability measures and L -Wasserstein distance. For more details consult [17] for example. ISTANCE FUNCTIONS ON CONVEX BODIES AND SYMPLECTIC TORIC MANIFOLDS 17
Let P ( R n ) be the set of all complete Borel probability measures on R n . Considerthe subset of P ( R n ) consisting of measures with finite quadratic moment, P ( R n ) := (cid:26) m ∈ P ( R n ) (cid:12)(cid:12)(cid:12)(cid:12) ∃ o ∈ R n , (cid:90) R n (cid:107) x − o (cid:107) m ( dx ) < ∞ (cid:27) . A.1.
Weak convergence and Prokhorov’s theorem.Definition A.1.1.
A sequence { m i } i in P ( R n ) converges weakly to m ∈ P ( R n )lim i →∞ (cid:90) R n f ( x ) m i ( dx ) = (cid:90) R n f ( x ) m ( dx )for any bounded continuous function f on R n . Theorem A.1.2.
For a sequence { m i } i in P ( R n ) and m ∈ P ( R n ) the followingsare equivalent. (1) { m i } i converges weakly to m . (2) For any open subset U in R n we have lim inf i →∞ m i ( U ) ≥ m ( U ) . (3) For any closed subset C in R n we have lim sup i →∞ m i ( C ) ≤ m ( C ) . (4) For any Borel subset A in R n with m ( A \ A ◦ ) = 0 we have lim i →∞ m i ( A ) = m ( A ) . Theorem A.1.3 (Prokhorov’s theorem) . A subset
K ⊂ P ( R n ) is relatively compactwith respect to the weak convergence topology if and only if for all (cid:15) > there existsa compact subset K ⊂ R n such that sup m ∈K m ( R n \ K ) < (cid:15). For a weak convergent sequence of probability measure the following is well-known. See [2] for example.
Theorem A.1.4. If { m i } i ⊂ P ( R n ) has a weak convergent limit m ∈ P ( R n ) , thenfor any x ∈ supp ( m ) there exists x i ∈ supp ( m i ) such that x i → x . A.2. L -Wasserstein distance of probability measures. For m, m (cid:48) ∈ P ( R n )let Cpl ( m, m (cid:48) ) be the set of all couplings between m and m (cid:48) . Namely Cpl ( m, m (cid:48) )is the set of measures ξ ∈ P ( R n × R n ) such that for any Borel subset A of R n itsatisfies (cid:40) ξ ( A × R n ) = m ( A ) ξ ( R n × A ) = m (cid:48) ( A ) . The L -Wasserstein distance between m, m (cid:48) ∈ P ( R m ) is defined by W ( m, m (cid:48) ) := inf (cid:40)(cid:18)(cid:90) R n × R n (cid:107) x − y (cid:107) ξ ( dx, dy ) (cid:19) / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ξ ∈ Cpl ( m, m (cid:48) ) (cid:41) . It is known that W is a metric on P ( R n ) and ( P ( R n ) , W ) is a complete sepa-rable metric space with the following properties. Theorem A.2.1.
For a sequence { m i } i in P ( R n ) and m ∈ P ( R n ) the followingsare equivalent. (1) W ( m i , m ) → ( i → ∞ ). (2) { m i } i converges weakly to m and lim R →∞ lim sup i →∞ (cid:90) R n \ B ( o,R ) (cid:107) x − o (cid:107) m i ( dx ) = 0 . A subset
K ⊂ P ( R n ) with this property is often called tight . (3) For any continuous function ϕ such that | ϕ ( x ) | ≤ C (1 + (cid:107) x − x (cid:107) ) forsome C > , x ∈ R n the following holds. lim i →∞ (cid:90) R n ϕ dm i = (cid:90) R n ϕ dm. Recall that if for m, m (cid:48) ∈ P ( R n ) there exists a Borel measurable map T : R n → R n such that T ∗ m = m (cid:48) and ( id × T ) ∗ m ∈ Opt ( m, m (cid:48) ), then we say that theMonge problem for m, m (cid:48) admits a solution and T is called a solution of the Mongeproblem. Theorem A.2.2.
For m, m (cid:48) ∈ P ( R n ) if m (cid:28) L n , then there is a solution of theMonge problem for m and m (cid:48) . The solution is unique in the following sense. Foranother solution S : R n → R n we have m ( { T (cid:54) = S } ) = 0 . Corollary A.2.3.
For m, m (cid:48) ∈ P ( R n ) with m (cid:28) L n and a sequence { m (cid:48) i } i in P ( R n ) which converges weakly to m (cid:48) , there exists a solution T : R n → R n of theMonge problem for m , m (cid:48) and a sequence { T i } i of solutions of the Monge problemfor m , m (cid:48) i with m ( { x ∈ R n | | T i ( x ) − T ( x ) | ≥ (cid:15) } ) → i → ∞ ) . Appendix B. Disintegration theorem
We use the following type of disintegration theorem. See [8, Theorem 16.10.1]for example.
Theorem B.0.1.
Let X and Y be complete separable metric spaces. Let m be a σ -finite Borel probability measure and f : X → Y a Borel measurable map. Supposethat the push forward f ∗ m is a σ -finite measure on Y . Then there exists a familyof probability measures { m y } y ∈ Y on X such that for each Borel subset A the map Y (cid:51) y (cid:55)→ m y ( A ) ∈ [0 , is Borel measurable and for each Borel measurable function ϕ on X we have (cid:90) X ϕ dm = (cid:90) Y (cid:18)(cid:90) X ϕ ( x ) m y ( dx ) (cid:19) f ∗ m ( dy ) . Moreover we have m y ( f − ( y )) = 1 ( y ∈ Y ( f ∗ m -a.e )) . The above family of measures { m y } y ∈ Y is called a disintegration for f : X → Y . References [1] M. Abreu,
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Japan Women’s University
E-mail address : [email protected] (Yu Kitabeppu) Kumamoto University
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