Maximal ball packings of symplectic-toric manifolds
Maximal ball packings of symplectic-toric manifolds
Alvaro Pelayo and Benjamin Schmidt
August 26, 2021
Abstract
Let (M, σ, ψ) be a symplectic-toric manifold of dimension at least four. This paper investigates the
so called symplectic ball packing problem in the toral equivariant setting. We show that the set of toric
symplectic ball packings of M admits the structure of a convex polytope. Previous work of the first
author shows that up to equivalence, only (CP1)2 and CP2 admit density one packings when n = 2 and
only CPn admits density one packings when n > 2. In contrast, we show that for a fixed n ≥ 2 and
each δ ∈ (0, 1), there are uncountably many inequivalent 2n-dimensional symplectic-toric manifolds
with a maximal toric packing of density δ. This result follows from a general analysis of how the den-
sities of maximal packings change while varying a given symplectic-toric manifold through a family of
symplectic-toric manifolds that are equivariantly diffeomorphic but not equivariantly symplectomorphic.
1 Introduction and Main Result
1.1 Motivation
What portion of a manifold can be filled by disjointly embedded balls? Answers to this ball packing question
depend considerably on the manifold being packed, the types of embeddings allowed, the number of balls
allowed, and the radii of balls allowed. In this paper, we consider a packing problem for a specific class of
compact symplectic manifolds. Throughout, we let (M2n, σ) denote a 2n-dimensional compact, connected,
smooth manifold equipped with a symplectic form σ. Before stating our main result, we discuss some of
the foundational results concerning ball packings of symplectic manifolds.
In [9], M. Gromov considered the problem of finding the supremum of the densities
νN (M, σ) = sup
N vol(Br)
volσ(M)
of packings of a symplectic manifold (M2n, σ) by a fixed number N of disjoint symplectic embeddings of
the standard open radius r symplectic ball Br ⊂ C
n. Therein, Gromov found obstructions to a full (also
http://arxiv.org/abs/0704.1036v2
called perfect) packing by too few balls of the same radius; specifically, he established that νN (B1) ≤ N/2
for each 1 < N < 2n. Motivated by techniques from algebraic geometry, D. McDuff and L. Polterovich
established a correspondence between symplectically embedded balls and symplectic blowing-up in [11].
They showed, for instance, that when N is a square, CP2 is fully packable by N disjoint symplectically
embedded balls. In contrast, they also established that νN (CP
2) < 1 for nonsquare 1 < N ≤ 8. In
[1], P. Biran demonstrated that for any symplectic four manifold (M4, σ) with rational cohomology class
[σ] ∈ H2(M, Q), νN(M, σ) = 1 for all sufficiently large N . We refer the reader to [2] for more in this
direction.
In this article, we consider the symplectic packing problem in a toral equivariant setting. Specifically,
we consider toric ball packings of symplectic-toric manifolds (see Definition 2.4). In this particular ball
packing problem, the Euler characteristic χ(M) gives an upper bound on the number of disjoint balls in
any possible packing. For this reason, we allow the balls in an equivariant packing to have varying radii as
opposed to the packings described in the previous paragraph where all balls have the same radius.
The study of toric ball packings of symplectic-toric manifolds was initiated by the first author in [12]
where the homotopy type of the space of equivariant and symplectic embeddings of a fixed ball is described,
and in [13] where the symplectic-toric manifolds admitting a full toric ball packing are classified. The
present work was motivated by the latter:
Theorem 1.1. [13, Thm. 1.7] A symplectic-toric manifold (M2n, σ, ψ) admits a full toric ball packing if
and only if there exists λ > 0 such that
• if n = 2, (M4, σ, ψ) is equivariantly symplectomorphic to either (CP2, λ · σFS) or a product (CP
1, λ · (σFS ⊕ σFS)) (where σFS denotes the Fubini-Study form and these manifolds are equipped
with the standard actions of T2), or
• if n = 1 or n > 2, (M2n, σ, ψ) is equivariantly symplectomorphic to (CPn, λ · σFS) (where σFS
denotes the Fubini-Study form and this manifold is equipped with the standard action of Tn).
1.2 Main Results
Our motivation for this paper was to understand when it is possible to list (up to equivalence) all symplectic-
-toric manifolds admitting a maximal toric ball packing of a specified density, such as in Theorem 1.1. As
it turns out, this is possible only when that density is one. To be more precise, let S2n denote the set
of equivalence classes of 2n-dimensional symplectic-toric manifolds (see Section 2 for definitions). The
maximal density function
Ω2n : S
2n → (0, 1]
associates to each equivalence class [(M2n, σ, ψ)] the largest density from all possible symplectic-toric
packings of M .
Theorem 1.1 classifies the set Ω−12n ({1}). In contrast with Theorem 1.1, we prove the following:
Theorem 1.2. Let S2n denote the set of equivalence classes of 2n-dimensional symplectic-toric manifolds
and let Ω2n : S
2n → (0, 1] be the maximal density function. Then Ω−1({x}) is uncountable for all x ∈
(0, 1).
Theorem 1.2 answers [13, Quest. 5.1]. The proof of Theorem 1.2 follows from the proof of the next
theorem, which asserts that there are uncountable families of equivariantly diffeomorphic symplectic toric
manifolds with the same maximal density that are not equivariantly symplectomorphic.
Theorem 1.3. Let S2n denote the set of equivalence classes of 2n-dimensional symplectic-toric manifolds
and let
Ω2n : S
2n → (0, 1]
be the maximal density function, n ≥ 2. Suppose that (M, σ, ψ) is a symplectic-toric manifold with Euler
characteristic χ(M) ≥ ⌊(n + 2)/2⌋ · ⌈(n + 2)/2⌉ + 1. Then for any ǫ > 0, there exists a constant c > 0
and a family F of equivariantly diffeomorphic symplectic-toric manifolds satisfying
• | volσ′(M
′)− volσ(M)| < ǫ for all (M
′, σ′, ψ′) ∈ F ,
• Ω−12n ({x}) ∩ F is uncountable for all x ∈ (δ − c, δ) or for all x ∈ (δ, δ + c).
To prove Theorem 1.3 we exploit the following:
Proposition 1.4. Let (M2n, σ, ψ) be a symplectic-toric manifold of dimension 2n ≥ 4. The set of symplectic-
-toric packings of (M2n, σ, ψ) has the structure of a convex polytope in RV , where V denotes the Euler
characteristic χ(M) of M . Moreover, (M2n, σ, ψ) admits finitely many maximal toric ball packings.
The proof of Theorem 1.3 essentially follows from the fact that there are certain linear perturbations
of a given symplectic-toric manifold along which the maximal density function is locally convex. The
proof of its local convexity follows from the Brunn-Minkowski inequality once an explicit description of
the maximal density function in terms of the polytope from Proposition 1.4 is given.
In the next section, we collect together preliminary material and reduce Theorem 1.3 to a proposition
concerning polytopes. In the last section we prove these propositions using basic convexity techniques.
Acknowledgements:
We are indebted to Professor Novik for bringing to our attention an explicit example showing that the
density function is surjective during the first author’s visit to the University of Washington, as well as for
providing us with the statement and proof of Lemma 2.16. The first author is grateful to her for stimulating
conversations and for hospitality at the University of Washington. Part of the work of the first author was
funded by Rackham Fellowships from the University of Michigan. His research was partially conducted
while visiting Oberlin College. The second author was partially funded by the Clay Mathematics Institute as
a Liftoff Fellow as well as by an NSF Postdoctoral Fellowship during the period this research was conducted.
2 Reduction to Convex Geometry
Throughout this section (M2n, σ) denotes a 2n-dimensional compact connected smooth manifold equipped
with a symplectic form. We let Tk ∼= (S1)k denote the k-dimensional torus and identify its Lie algebra t
with Rk. This identification is not unique and throughout we make the convention that the identification
comes from the isomorphism Lie(S1) ∼= R given by ∂
7→ 1/2. Using the standard inner product on Rk,
we identify the dual space t∗ with t.
Definition 2.1. A σ-preserving action ψ : Tk ×M → M of a k-dimensional torus is Hamiltonian if for
each ξ ∈ t there exists a smooth function µξ : M → R such that iξMσ = dµξ, and the map
t → C∞(M, R), ξ 7→ µξ,
is a Lie algebra homomorphism. Here, ξM is the vector field on M infinitesimally generating the one
parameter action coming from ξ and the Lie algebra structure on C∞(M, R) is given by the Poisson bracket.
It follows from Definition 2.1 that a σ-preserving action ψ : Tk ×M → M of a k-dimensional torus is
Hamiltonian if and only if there exists a momentum map µM :M → t∗ satisfying Hamilton’s equation
iξMσ = d〈µ
M , ξ〉,
for all ξ ∈ t. Note that a momentum map is well defined up to translation by an element of t∗. Nevertheless,
we will ignore this ambiguity and refer to the momentum map. It is well known that if (M2n, σ) admits an
effective and Hamiltonian action of Tk, then k ≤ n (see for instance [3, Theorem 27.3]). The maximal case,
usually referred to as a symplectic-toric manifold or Delzant manifold, is a triple (M2n, σ, ψ) consisting
of a compact connected symplectic manifold (M, σ) equipped with an effective and Hamiltonian action
ψ : Tn ×M →M .
Example 2.2. Equip the open radius r ball Br ⊂ Cn with the standard symplectic form σ0 =
j dzj∧dzj .
The action Rot : Tn×Br → Br of T
n given by (θ1, . . . , θn)·(z1, . . . , zn) = (θ1 z1, . . . , θn zn) is Hamiltonian.
Its momentum map µBr has components µBrk = |zk|
2. Its image, which we shall denote by ∆n(r), is given
∆n(r) = ConvHull(0, r2 e1, . . . , r
2 en) \ ConvHull(r
2 e1, . . . , r
2 en),
where {ei}
i=1 is the standard basis of R
n. When the dimension n is clear from the context, we shall write
∆(r) = ∆n(r). ⊘
The next two definitions define the packings considered in this paper. These definitions first appeared in
[13] in a slightly different but equivalent form.
Definition 2.3. Let (M2n, σ, ψ) be a symplectic-toric manifold, let Λ ∈ Aut(Tn) and let r > 0. A subset
B ⊂ M is said to be a Λ-equivariantly embedded symplectic ball of radius r if there exists a symplectic
embedding f : Br →M with image B and such that the following diagram commutes:
Tn × Br
Λ×f //
Tn ×M
f // M
We say that the Λ-equivariantly embedded symplectic ball B has center f(0) ∈M .
We shall say that a subset B′ ⊂ M is an equivariantly embedded symplectic ball of radius r′ if there
exists Λ′ ∈ Aut(Tn) such that B′ is a Λ′-equivariantly embedded symplectic ball of radius r′ (although this
is a slight abuse of the standard use of the word “equivariantly”). ⊘
We define the symplectic volume of a subset A ⊂M by volσ(A) :=
Definition 2.4. Let (M2n, σ, ψ) be a symplectic-toric manifold. A toric ball packing of M is a disjoint
union P :=
α∈ABα of equivariantly embedded symplectic balls Bα (of possibly varying radii) in M . The
density Ω(P) of a packing P is defined by Ω(P) := volσ(P)/ volσ(M). The density Ω(M
2n, σ, ψ) of a
symplectic-toric manifold (M2n, σ, ψ) is defined by
Ω(M, σ, ψ) := sup{Ω(P) | P is a toric packing ofM}.
A packing achieving this density is said to be a maximal density packing. If in addition, this density is one,
then (M2n, σ, ψ) is said to admit a full or perfect toric ball packing. ⊘
For a symplectic-toric manifold (M2n, σ, ψ), the number of fixed points of ψ is known to coincide with
the Euler characteristic χ(M) (see e.g. [5]). It follows that a toric ball packing P consists of at most χ(M)
disjoint equivariant balls. By a well known theorem of Atiyah and Guillemin-Sternberg, the image of the
momentum map of a toral Hamiltonian action is the convex hull of the images of the fixed points (see for
instance [3, Theorem 27.1]). The images of momentum maps for symplectic-toric manifolds are a particular
class of polytopes. Recall that an n-dimensional polytope is simple if there are precisely n edges meeting
at each one of its vertices.
Definition 2.5. A simple n-dimensional convex polytope ∆ ⊂ Rn is said to be Delzant if for each vertex
v, the edges meeting at v are all of the form v + ti ui where ti > 0 and {u1, . . . , un} define a basis of the
Z-module Zn. ⊘
A polytope is describable as the intersection of closed half-spaces
{x ∈ Rn | 〈x, ui〉 ≥ λi},
where the vector ui is an inward pointing normal vector to the i
th facet of ∆ and each λi is a real scalar. In
this notation, the polytope ∆ is Delzant if and only if there are precisely n facets incident to each vertex of
∆ and the inward pointing normals to these facets u1, . . . , un can be chosen to be a Z-basis of Z
Given a symplectic-toric manifold M , the symplectic-toric manifolds obtained from M by scaling the
symplectic form, changing the time parameter in the acting torus by an automorphism, and any others that
are equivariantly symplectomorphic to one of these will clearly have the same maximal density. Therefore,
for the purpose of this paper we will say that two symplectic-toric manifolds (M1, σ1, ψ1) and (M2, σ2, ψ2)
are equivalent if there exists an automorphism Λ ∈ Aut(Tn) ∼= GL(n, Z), a positive number λ > 0, and a
symplectomorphism h : (M1, σ1) → (M2, λ · σ2) such that the following diagram commutes:
Tn ×M1
Λ×h //
Tn ×M2
h // M2
We recall the result of Delzant [4]:
Theorem 2.6. Suppose that (M1, σ1, ψ1) and (M2, σ2, ψ2) are two 2n-dimensional symplectic-toric man-
ifolds with momentum maps µM1 and µM2 respectively. Then there exists a (ψ1, ψ2)-equivariant symplecto-
morphism h : (M1, σ1) → (M2, σ2) such that µ
M1 = µM2 ◦ h if and only if µM1(M1) = µ
M2(M2).
In view of this theorem, there is a natural equivalence relation one can put on the set of Delzant polytopes
so that momentum maps will induce a bijective correspondence between equivalence classes of symplectic-
-toric manifolds as above and equivalence classes of Delzant polytopes. To be more specific, first note that
scaling Rn leaves invariant the Delzant polytopes. Define two Delzant polytopes ∆1,∆2 ⊂ R
n to be in the
same projective class if there exists λ > 0 such that λ∆1 = ∆2. The group AGL(n, Z) consisting of affine
transformations of Rn with linear part in GL(n, Z) acts on the set of projective classes of Delzant polytopes
in Rn. For the purpose of this paper we say that two Delzant polytopes ∆1 and ∆2 are equivalent if the
projective classes of ∆1 and ∆2 are in the same AGL(n, Z) orbit. By applying Theorem 2.6, it is standard
to show that there is a bijective correspondence between equivalence classes of symplectic-toric manifolds
and equivalence classes of Delzant polytopes as defined here.
We exploit this correspondence in order to reduce Theorem 1.3 and Proposition 1.4 to propositions
concerning packing Delzant polytopes. The next few definitions are translations of the above definitions
into the appropriate definitions concerning Delzant polytopes. Again, they first appeared in [13] in a slightly
different but equivalent form.
Definition 2.7. Let ∆ be a Delzant polytope. A subset Σ ⊂ ∆ is said to be an admissible simplex of radius
r with center at the vertex v ∈ ∆ if Σ is the image of ∆(r1/2) by an element of AGL(n, Z) which takes the
origin to v and the edges of ∆(r1/2) to the edges of ∆ meeting v. For a vertex v ∈ ∆, we put
rv := max{r > 0 | ∃ an admissible simplex of radius r with center v}. ⊘ (1)
Remark 2.8. In view of Example 2.2, the simplex ∆(r1/2) may be identified with the set obtained by
removing from ConvHull(0, re1, . . . , ren) the facet not containing the origin. For this reason we say that
AGL(n, Z) images of ∆(r1/2) as in the above definition have radius r instead of radius r1/2. ⊘.
We denote the Euclidean volume of a subset A ⊂ ∆ by voleuc(A).
Definition 2.9. Let ∆ be a Delzant polytope. An admissible packing of ∆ is a disjoint union P :=
α∈A Σα
of admissible simplices (of possibly varying radii) in ∆. The density Ω(P) of a packing P is defined by
Ω(P) := voleuc(P)/ voleuc(∆). The density Ω(∆) of a Delzant polytope ∆ is defined by
Ω(∆) := sup{Ω(P) | P is an admissible packing of∆}.
A packing achieving this density is said to be a maximal density packing. If in addition, this density is one,
then ∆ is said to admit a full or perfect packing. ⊘
The next lemma shows that admissible simplices in ∆ are parametrized by their centers and radii. The
rational or SL(n, Z)-length of an interval I ⊂ Rn with rational slope is the unique number l := lengthQ(I)
such that I is AGL(n, Z)-congruent to an interval of length l on a coordinate axis. For a vertex v in a
Delzant polytope, we denote the n edges leaving v by e1v, . . . , e
v . By Definition 2.5, each e
v is of the form
v+ tiv u
v with t
v > 0 and {u
i=1 defining a Z-basis of Z
n. In this notation, we have that lengthQ(e
v) = t
Lemma 2.10. Let ∆ be a Delzant polytope. Then for each vertex v ∈ ∆,
rv = min{lengthQ(e
v), . . . , lengthQ(e
v )}.
There is an admissible simplex Σ(v, r) of radius r with center v if and only if 0 ≤ r ≤ rv. Moreover this
admissible simplex is unique and voleuc(Σ(v, r)) = r
n/n!.
Proof. We first argue the uniqueness of an admissible simplex Σ of radius r and center v, assuming its
existence. Suppose Σ1 and Σ2 were two such. By definition, there exist two affine transformationsA1, A2 ∈
AGL(n, Z) satisfying A1(∆(r
1/2)) = Σ1 and A2(∆(r
1/2)) = Σ2. As both Σi are centered at v, both Ai
have translational part given by v. Write Ai(·) = Λi(·)+ v with Λi ∈ GL(n, Z). By the Delzant property of
∆, c.f. Definition 2.5, the automorphisms Λ1 and Λ2 both take the standard basis {ei}
i=1 of Z
n bijectively
onto the basis {uiv}
i=1 of Z
n as unordered sets. Therefore Λ−11 Λ2 leaves invariant the standard basis of Z
as an unordered set and hence leaves ∆(r1/2) invariant as a set. It follows that Σ1 = Σ2. Next we argue
existence.
As above, write eiv = v + t
v with {u
i=1 forming a Z-basis of Z
n. Let Λ ∈ GL(n, Z) be the
automorphism of Zn defined by Λ(ei) = u
v, with {ei}
i=1 the standard basis of Z
n. The affine transfor-
mation A : Rn → Rn defined by A(x) = Λ(x) + v satisfies A(t ei) = t u
v + v for each t > 0. There-
fore, A(∆(r1/2)) defines an admissible simplex for ∆ with center v if and only if r ≤ min{t1v, . . . , t
min{lengthQ(e
v), . . . , lengthQ(e
v )}, concluding the proof of existence.
Finally, for each 0 ≤ r ≤ rv, let Σ(v, r) be the unique admissible simplex with radius r and center v
guaranteed by the previous two paragraphs. Since elements in AGL(n, Z) preserve Euclidean volume, we
have voleuc(Σ(v, r)) = voleuc(∆(r
1/2)) = rn/n!, completing the proof.
Lemmas 2.13 and 2.14 below reduce the problem of analyzing the densities of toric ball packings of
symplectic-toric manifolds to analyzing the densities of admissible packings of Delzant polytopes. These
lemmas appear in [12] and [13] respectively, in slightly different form. We include the arguments here for
the reader’s convenience. We first recall some definitions and a result due to Y. Karshon and S. Tolman.
Let (N2n, σN) denote a connected (and not necessarily compact) symplectic manifold equipped with
an effective Hamiltonian action of Tn with momentum map µN : N → t∗. Suppose that T ⊂ t∗ is an open
and convex subset containing µN(N) with the property that µN : N → T is a proper map. The quadruple
(N, σN , µN , T ) is said to be a proper Hamiltonian Tn-manifold. For a subgroup K ⊂ Tn, denote the
fixed point set for the Tn action on N by NK . A proper Hamiltonian Tn-manifold (N, σN , µN , T ) is said
to be centered at α ∈ T provided that α is in the momentum map image of every component of NK for
all subgroups K ⊂ Tn. For a fixed point p ∈ NT
, there exist isotropy weights η1, . . . , ηn ∈ t
∗ such that
the induced linear symplectic action on Tp(N) is isomorphic to the action on (C
n, σ0) generated by the
momentum map
(x) = µN(p) +
|zj |
2ηj .
For a symplectic-toric manifold M with fixed point p ∈ MT
, and corresponding vertex v := µ(p) ∈ ∆,
and edges eiv = v + t
v (i = 1, . . . , n) emanating from v , the Z-basis {u
i=1 of Z
n coincides with the
set of isotropy weights at p.
Proposition 2.11. [10, Prop. 2.8] Let (N2n, σN , µN , T ) be a proper Hamiltonian Tn-manifold. Assume
that N is centered about α ∈ T and that (µN)−1({α}) consists of a single fixed point p. Then N is
equivariantly symplectormorphic to
{z ∈ Cn |α+
|zj |
2ηj ∈ T },
where η1, . . . , ηn ∈ t
∗ are the isotropy weights at p.
Remark 2.12. The reader who consults [10] will find an additional factor of π in the definition of isotropy
weights and in the statement of their Proposition 2.8. This factor does not appear in the present work
because of the particular identification chosen between t and Rn. ⊘
Lemma 2.13. Let (M2n, σ, ψ) be a symplectic-toric manifold with momentum map µM : M → t∗ and
associated Delzant polytope ∆ := µM(M). Let B ⊂ M be an equivariantly embedded symplectic ball of
radius r and center p ∈ M . Then µM(B) is an admissible simplex of radius r2 in ∆ with center µM(p).
Conversely, if Σ ⊂ ∆ is an admissible simplex of radius r, then there exists an equivariantly embedded
symplectic ball B ⊂M of radius r1/2 satisfying µM(B) = Σ.
Proof. First suppose that B ⊂ M is an equivariantly embedded ball of radius r. By definition, there is an
automorphism Λ ∈ Aut(Tn) ∼= GL(n, Z) and a symplectic embedding f : (Br, σ0) → (M, σ) with image
B such that the following diagram commutes:
Tn × Br
Λ×f //
Tn ×M
f // M
We denote by ξBr and ξM the vector fields on Br and M infinitesimally generating the actions of the one
paramater group coming from ξ ∈ t. Fix z ∈ Br and a tangent vector v ∈ TzBr.
Note that from the definition of the momentum maps µM and µBr , commutativity of the above diagram,
and the fact that f ∗σ = σ0, we have the following sequence of equalities:
〈dµBrz (v), ξ〉µBr (z) = σ0(v, ξBr)z
= σ(dfz(v), dfz(ξBr))f(z)
= σ(dfz(v), (Λ ξ)M)f(z)
= 〈dµMf(z)(dfz(v)), Λ ξ〉µM (f(z))
= 〈Λt dµMf(z)(dfz(v)), ξ〉µBr (z). (2)
By equation (2) and the chain rule we obtain that for all z ∈ Br and v ∈ TzBr,
dµBrz (v) = d(Λ
t ◦ µM ◦ f)z(v). (3)
As Br is connected, (3) implies that there exists x
′ ∈ Rn such that µBr + x′ = Λt ◦ µM ◦ f as maps
Br → R
n. Letting x = (Λt)−1(x′) yields commutativity of the following diagram:
(Λt)−1+x
// ∆M
. (4)
It follows from commutativity of this diagram that x = µM(f(0)) = µM(p) so that µ(B) = µ(f(Br)) is an
admissible simplex of radius r2 and center µM(p), completing the proof of the first statement.
Next suppose that Σ ⊂ ∆ is an admissible simplex of radius r. By applying a translation, we may
assume that Σ is centered at the origin. Identify Σ with the set
ConvHull(0, r η1, . . . , r ηn) \ ConvHull(r η1, . . . , r ηn).
Let T ⊂ t∗ be the the unique open half space of t∗ containing Σ with bounding hyperplane containing
ConvHull(r η1, . . . r ηn). Denote by σ
N , µN the restrictions of the symplectic form σ, and of the momen-
tum map µM , to the open submanifold N := (µM)−1(Σ) ⊂ M . The quadruple (N, σN , µN , T ) is a
proper Hamiltonian Tn manifold centered at 0 ∈ T . It now follows from Proposition 2.11, that (N, σN ) is
equivariantly symplectomorphic to
{z ∈ Cn |
|zj |
2ηj ∈ T } = Br1/2 .
In other words, the set N ⊂ M is an equivariantly embedded symplectic ball of radius r1/2 satisfying
µM(N) = Σ (c.f. [13, Lem 2.3] for an explicit verification).
Lemma 2.14. Let (M2n, σ, ψ) be a symplectic-toric manifold with momentum map µM :M → Rn and as-
sociated Delzant polytope ∆ := µM(M). Then for each toric ball packing P of M , µM(P) is an admissible
packing of ∆ satisfying Ω(P) = Ω(µM(P)). Moreover, given an admissible packing Q of ∆, there exists a
toric ball packing P of M satisfying µM(P) = Q.
Proof. Let P be a toric ball packing ofM . For each equivariant symplectic ball B in the packing P , µM(B)
is an admissible simplex in ∆ by the previous lemma. Since the fibers of the momentum map µM :M → ∆
are connected (see for instance [3, Theorem 27.1]), disjoint equivariant symplectic balls in the packing P
are sent to disjoint admissible simplices in ∆. Hence, µM(P) is a toric packing of ∆. By the Duistermaat-
-Heckman Theorem (see for instance [3, Theorem 30.3]) the push forward of symplectic volume under the
momentum map satisfies µM∗ (volM) = K(n) voleuc, where K(n) > 0 is a dimensional constant. It follows
that Ω(P) = Ω(µM(P)). It remains to argue that given an admissible packing Q of ∆, there exists a toric
ball packing P of M satisfying µM(P) = Q. By the Lemma 2.13, for each admissible simplex Σ there
exists an equivariant symplectic ball B ⊂ M with µM(B) = Σ. Choosing one such equivariant symplectic
ball for each admissible simplex in Q defines a disjoint collection P of equivariant symplectic balls mapping
onto Q under the momentum map. Hence P is a toric packing of M satisfying µM(P) = Q.
Before concluding this section by reformulating our main results in terms of polytopes, we must first
introduce the complete regular n–dimensional fan associated to an n-dimensional Delzant polytope ∆, as
in [4, Sec. 5]. As above, write
{x ∈ Rn | 〈x, ui〉 ≥ λi},
where F is the number of facets of ∆ and ui is the unique primitive integral normal vector to the i
th facet.
For each face ∆′ of ∆ of codimension k, there is a unique multi-index I∆′ of length k, I∆′ = {i1, . . . , ik},
1 ≤ i1 < . . . < ik ≤ F , such that
∆′ = {x ∈ Rn | 〈x, uj〉 = λj , ∀j ∈ I∆′}.
Letting σ∆′ denote the cone in R
n generated by the vectors {uj | j ∈ I∆′}, the complete regular n-
-dimensional fan associated to ∆ is given by {σ∆′ |∆
′ is a face of∆}. For our purposes here, we state
the well known fact that if two Delzant polytopes have the same associated fan, their associated symplectic-
-toric manifolds are equivariantly diffeomorphic. This is a standard fact that follows from the construction
of a symplectic-toric manifold starting from a given Delzant polytope.
It now follows from Lemma 2.13 and Lemma 2.14 that proving Theorem 1.2 is reduced to establishing
the following proposition:
Proposition 2.15. Let Dn denote the set of equivalence classes of n-dimensional Delzant polytopes and
Ω : Dn → (0, 1]
be the maximal density function, n ≥ 2. Then Ω−1({x}) is uncountable for all x ∈ (0, 1).
Recall that the number of fixed points of the Tn-action on M equals the Euler Characteristic of M and
the number of vertices of the momentum polytope µ(M), c.f. [5]. We are grateful to Professor Novik for
the following observation:
Lemma 2.16. An n-dimensional Delzant polytope ∆ with at least ⌊(n+2)/2⌋ · ⌈(n+2)/2⌉+1 vertices has
at least n + 3 facets. Moreover, this bound is sharp in the sense that there exists an n-dimensional Delzant
polytope with ⌊(n+ 2)/2⌋ · ⌈(n+ 2)/2⌉ vertices and n+ 2 facets.
Proof. Indeed, to see that a fewer number of vertices is not enough, let ∆ be the direct product of two
regular simplexes, one of dimension ⌊n/2⌋ and another one of dimension ⌈n/2⌉. Their product is an n-
dimensional Delzant polytope that has ⌊(n+ 2)/2⌋ · ⌈(n+ 2)/2⌉ vertices and only (n+ 2) facets. It is well
known, see e.g. [7, pp. 98-100] for the proof of (the dual) statement, that ∆ has the maximal number of
vertices amongst all simple n-polytopes with n+ 2 facets.
It then follows from Lemma 2.13, Lemma 2.14 and Lemma 2.16 that proving Theorem 1.3 is reduced
to establishing the following proposition:
Proposition 2.17. Let Dn denote the set of equivalence classes of n-dimensional Delzant polytopes and
Ω : Dn → (0, 1]
be the maximal density function, n ≥ 2. Suppose that ∆ is a Delzant polytope having at least n + 3 facets
and let Ω(∆) := δ ∈ (0, 1). Then for any ǫ > 0, there exists a constant c > 0 and a family F of Delzant
polytopes satisfying
• the polytopes in F determine a common fan,
• | voleuc(∆
′)− voleuc(∆)| < ǫ for all ∆
′ ∈ F ,
• Ω−1({x}) ∩ F is uncountable for all x ∈ (δ − c, δ) or for all x ∈ (δ, δ + c).
Similarly we have reduced proving Proposition 1.4 to showing:
Proposition 2.18. Let ∆ ⊂ Rn be a Delzant polytope. The set of admissible packings of ∆ has the structure
of a convex polytope in RV , where V is the number of vertices of ∆. Moreover, ∆ admits finitely many
maximal density packings.
We prove these propositions in the next section.
3 Proofs of propositions 2.15, 2.17, and 2.18.
In this section, we prove Propositions 2.15, 2.17, and 2.18 using convexity arguments. First we recall some
preliminary notions. For a set A ⊂ Rn, denote the closure of A by A. Now suppose that A is a convex set.
A function f : A→ R is said to be convex if for all x1, x2 ∈ A and t ∈ (0, 1),
f((1− t) x1 + t x2) ≤ (1− t) f(x1) + t f(x2)
and strictly convex if the inequality is always strict. Similarly, the function f is said to be concave if for all
x1, x2 ∈ A and t ∈ (0, 1),
f((1− t) x1 + t x2) ≥ (1− t) f(x1) + t f(x2)
and strictly concave if the inequality is always strict. It follows from the definitions that if f is a convex
function on A and g is a positive concave function on A, then f/g is a convex function which is strict if one
of f or g is strict. Moreover, if f1, . . . , fk are convex functions onA then so is the function max{f1, . . . , fk}.
We let C(Rn) denote the space of compact convex subsets of Rn and endow it with the Hausdorff metric
dH given by
dH(A, B) := inf{ǫ > 0 |A ⊂ Nǫ(B) andB ⊂ Nǫ(A)},
where Nǫ(X) denotes the open ǫ-neighborhood of a subset X ⊂ R
n. A compact convex set A ∈ C(Rn)
with non-empty interior is said to be a convex body. If λ > 0 and A and B are convex bodies then so are the
λA := {λ a | a ∈ A} A+B = {a+ b | a ∈ A and b ∈ B}.
Subsets A, B ⊂ Rn are said to be homothetic if there exists v ∈ Rn and λ > 0 such that λA + {v} = B.
We recall the celebrated Brunn-Minkowski inequality (see [8] for a detailed survey):
Theorem 3.1 (Brunn-Minkowski). Let A, B be convex bodies in Rn and 0 < λ < 1. Then
vol1/neuc ((1− λ)A+ λB) ≥ (1− λ) vol
euc (A) + λ vol
euc (B),
with equality if and only if A and B are homothetic.
In the remainder of this section, we let ∆n =
i=1{x ∈ R
n | 〈x, ui〉 ≥ λi} denote an n-dimensional
Delzant polytope with F facets and V vertices. We enumerate the vertices v1, v2, . . . , vV and facets
F1, . . . ,FF . When two vertices vj and vk are adjacent in ∆, we denote their common edge by ej,k. By
the Delzant property, each vertex vi is the unique intersection point of n facets,
F jvi = {x ∈ R
n | 〈x, uvi〉 = λvi} ∩∆, j = 1, . . . , n,
where the inward normal vectors ujvi to the j
th facet F jvi collectively define a Z-basis of Z
n. In particular, ∆
is simple and there are precisely n edges ejvi , j = 1, . . . n leaving each vertex vi of ∆. We shall denote the
set of admissible packings of ∆ by AP(∆).
Proposition 2.18. Let ∆ ⊂ Rn be a Delzant polytope. The set AP(∆) of admissible packings of ∆ has the
structure of a polytope in RV , where V is the number of vertices of ∆. Moreover, ∆ admits finitely many
maximal toric ball packings.
Proof. Each packing Q ∈ AP(∆) consists of a disjoint union
i=1Σ(vi, Ri(Q)) of admissible simplices
Σ(vi, Ri(Q)) centered at the vertex vi with (possibly zero) radius Ri(Q). Define the map
R : AP(∆) → ΠVi=1[0, rvi], Q 7→ (R1(Q), . . . , RV (Q)).
By Lemma 2.10, the map R is injective so that we can identify AP(∆) with its image in ΠVi=1[0, rvi ].
Note that R is not surjective as the admissible simplices with given radii (x1, . . . , xV ) ∈ Π
i=1[0, r
vi] will
not in general be disjoint. We must argue that the image set of R is precisely the solution set to finitely
many linear inequalities.
As a first step, we give a criterion for admissible simplices to be disjoint in terms of their intersections
along the edges of ∆. To this end, let F denote a finite family of admissible simplices with pairwise distinct
centers and fix an admissible simplex Σ from the family. Let v denote the center of Σ.
Disjointness Condition: For Σ to be disjoint from the rest of the familyF , it is a necessary and sufficient
condition that its closure Σ intersects the closure of the other admissible simplices in the family in at most
one point in each of the edges ejv, j = 1, . . . , n.
To see that the above disjointness condition is necessary, suppose that the closure of another admissible
simplex in F , say Σ′, intersects Σ in more than a single point along some edge ejv. Then Σ ∩ Σ
′ ∩ ejv is a
convex subset of ejv with at least two points and is therefore a closed subsegment e ⊂ e
i with nonempty
interior. The interior of e is contained in both Σ and Σ′, whence Σ ∩ Σ′ 6= ∅. Next we argue that the
above condition is sufficient. To see this, let Σ′ be another admissible simplex in the family F . Denote
by v′ ∈ ∆ the center of Σ′ and let xjv′ denote the unique point in the set Σ
′ ∩ e
v′ \ Σ
′ ∩ e
v′ for each
j = 1, . . . , n. If the above condition holds, then {v′, x1v′ , . . . , x
v′} ⊂ ∆ \Σ. But since ∆ \Σ is a convex set
and Σ′ = ConvHull(v′, x1v′ , . . . , x
v′), it follows that Σ
′ ⊂ ∆ \ Σ, concluding the proof of sufficiency.
For distinct vertices vi, vj ∈ {v1, . . . , vV }, define Li,j by
Li,j =
lengthQ(ei,j) if vi and vj are adjacent
rvi + rvj otherwise
It follows from (1) and Lemma 2.10 and the preceding disjointness condition that a point (x1, . . . , xV ) ∈
ΠVi=1[0, rvi] lies in the image of R if and only if the equations
xi + xj ≤ Li,j ,
hold for all i 6= j ∈ {1, . . . , V }. Therefore,
R(AP(∆)) =
i 6=j∈{1,...,V }
{(x1, . . . , xV ) ∈ R
≥0 | xi + xj ≤ Li,j}. (5)
By Lemma 2.10, the density function in the coordinates of RV≥0 is given by
Ω((x1, . . . , xV )) = (Σ
i )/(n! voleuc(∆)).
As n > 1, Ω is a strictly convex function on AP(∆) so that its maximum value can only be obtained at its
vertices, establishing the last part of the proposition.
In view of the injectivity of R, we will henceforth identify AP(∆) with the polytope R(AP(∆)).
Before proving Proposition 2.17, we need to set up some additional notation and prove a lemma. A
natural way to perturb the Delzant polytope ∆ is by perturbing its defining linear equations. Define the map
F → C(Rn), s = (s1, . . . , sF ) 7→ ∆s (6)
∆s :=
{x ∈ Rn | 〈x, ui〉 ≥ λi + s
This map is continuous and has image in the set of polytopes with not more than F facets (we regard the
empty set as a polytope). For small enough r > 0, the polytopes ∆s with s ∈ B(0, r) ⊂ R
F still have
F facets, are Delzant, and all determine the same fan. Let B∆ denote the largest open ball centered at the
origin with these three properties and let Dn denote the set of equivalence classes of Delzant polytopes. The
map B∆ → D
n induced by (6) induces an equivalence relation on B∆ by declaring points in the fibers to
be equivalent. We denote the set of equivalence classes by B̂∆ and endow it with the quotient topology. As
GL(n, Z) is discrete, the dimension of a fixed equivalence class of Delzant polytopes is n+1, the dimension
of the group of homotheties of Rn. It follows that the dimension of B̂∆ satisfies dim(B̂∆) ≥ F − (n+ 1).
Define the function
Ω : B∆ → (0, 1]
by Ω(s) := Ω(∆s), and for s1, s2 ∈ B∆, define the function
Ωs1, s2 : [0, 1] → (0, 1]
by Ωs1, s2(t) := Ω((1 − t) s1 + t s2) = Ω(∆(1−t) s1+t s2).
Lemma 3.2. The function Ω : B∆ → (0, 1] is continuous. For distinct s1, s2 ∈ B∆, there exists a suitably
small ǫ(s1, s2) > 0 such that the restriction Ω
s1, s2
|[0, ǫ] : [0, ǫ] → (0, 1] is convex. Furthermore, if ∆s1 and
∆s2 are not homothetic, then Ω
s1, s2
|[0, ǫ] is strictly convex.
Proof. Define || · ||n : R
≥0 → R by ||(x1, . . . , xV )||n = (
i=1 x
1/n. It is well known that || · ||n is a strictly
convex function. For s ∈ B∆, we have that
Ω1/n(s) =
max || · ||n|AP(∆s)
(n!)1/n vol1/neuc (∆s)
Each of the vertices v1, . . . , vV of ∆ are defined as the unique solution to the linear system:
〈ujvi, vi〉 = λ
j = 1, . . . n.
Similarly, ∆s has V vertices v1(s), . . . , vV (s), each defined by the linear system:
〈ujvi, vi(s)〉 = λ
+ sjvi j = 1, . . . n.
Hence, the vertices define affine maps vi : B∆ → R
n. It follows that the edges of ∆s and their rational
lengths also vary linearly with s ∈ B∆ and that, as we have already remarked, s 7→ ∆s defines a continuous
map B∆ → C(R
By the proof of Proposition 2.18, c.f. expression (5),
AP(∆s) =
i 6=j∈{1,...,V }
{(x1, . . . , xV ) ∈ R
≥0 | xi + xj ≤ Li,j(s)},
where
Li,j(s) =
lengthQ(ei,j(s)) if vi(s) and vj(s) are adjacent;
rvi(s) + rvj (s) otherwise,
rvi(s) = min{lengthQ(e
(s)), . . . , lengthQ(e
(s))}.
Hence, the defining equations of AP(∆s) vary continuously with s ∈ B∆ so that s 7→ AP(∆s) defines a
continuous map B∆ → C(R
≥0). Since vol
1/n and max || · ||n define continuous maps on C(R
n) and C(RV≥0),
it follows that Ω is continuous.
Now fix s1, s2 ∈ B∆ and t ∈ (0, 1). We first claim that
∆(1−t) s1+t s2 = (1− t)∆s1 + t∆s2.
Note that
∆(1−t) s1+t s2 = ConvHull(v1((1− t) s1 + t s2), . . . , vV ((1− t) s1 + t s2)).
Since vi((1− t) s1 + t s2) = (1− t) vi(s1) + t vi(s2),
{v1((1− t) s1 + t s2), . . . , vV ((1− t) s1 + t s2)} ⊂ (1− t)∆s1 + t∆s2 ,
whence
∆(1−t) s1+t s2 ⊂ (1− t)∆s1 + t∆s2 .
Now consider the (n+ 1)-dimensional polytope
∆(s1, s2) :=
{(x, t) ∈ Rn × [0, 1] | 〈x, ui〉 ≥ λi + (1− t) s1 + t s2}.
Let Ht = ∆(s1, s2) ∩ {xn+1 = t} and note that Ht is naturally identified with ∆(1−t) s1+t s2 . Now, if
(x, 0) ∈ H0, (y, 1) ∈ H1, and t ∈ (0, 1), then (1 − t) (x, 0) + t (y, 1) ∈ Ht, concluding the proof of the
claim.
By Theorem 3.1, the map [0, 1] → R given by t 7→ vol1/neuc (∆(1−t) s1+t s2) is concave and strictly concave
if and only if ∆s1 and ∆s2 are not homothetic. To conclude the proof of the Lemma, we must argue that there
is a suitably small ǫ = ǫ(s1, s2) > 0 such that the map [0, 1] → R given by t 7→ max || · ||n|AP(∆(1−t) s1+t s2)
is convex when restricted to the interval [0, ǫ). Let p1, p2, . . . , pk ∈ R
V denote the vertices of AP(∆s1).
It follows from the description above that for t sufficiently small AP(∆(1−t) s1+t s2) also has k vertices
v1(t), . . . , vk(t) ∈ R
V and that the maps t 7→ vi(t) define (possibly constant) line segments in R
V . By
convexity of || · ||n,
max || · ||n|AP(∆(1−t) s1+t s2 ) = max{||v1(t)||n, . . . , ||vk(t)||n},
and the result follows.
We are now ready to prove:
Proposition 2.17. Let Dn denote the set of equivalence classes of n-dimensional Delzant polytopes and
Ω : Dn → (0, 1]
be the maximal density function, n ≥ 2. Suppose that ∆ is a Delzant polytope having at least n + 3 facets
and let Ω(∆) := δ ∈ (0, 1). Then for any ǫ > 0, there exists a constant c > 0 and a family F of Delzant
polytopes satisfying
• the polytopes from F determine a common fan,
• | voleuc(∆
′)− voleuc(∆)| < ǫ for all ∆
′ ∈ F ,
• Ω−1({x}) ∩ F is uncountable for all x ∈ (δ − c, δ) or for all x ∈ (δ, δ + c).
Proof. Let ∆ be a Delzant polytope with with at least n + 3 facets and with Ω(∆) = δ and let ǫ > 0. By
continuity of the volume function with respect to the Hausdorff metric, there is a suitably small connected
neighborhood N ⊂ B∆ of the origin for which | voleuc(∆s)− voleuc(∆)| < ǫ for each s ∈ N . We define the
desired family F by
F = {∆s | s ∈ N}
and remark that by construction all ∆′ ∈ F determine the same fan. Therefore, it remains to establish the
third item of the proposition.
As dim(N) ≥ n + 3 and since the space of homotheties of Rn has dimension n + 1, there exists
s ∈ N\{0} such that ∆s is not homothetic to∆. By Lemma 3.2, there exists ǫ
′ > 0 such that Ω
0, s : [0, ǫ
[0, 1) is strictly convex and therefore Ω: N → (0, 1] is not the constant map. Since Ω is continuous and N
is connected, there exists c > 0, such that (δ, δ + c) ⊂ Ω(N) or (δ − c, δ) ⊂ Ω(N).
Suppose that (δ − c, δ) ⊂ Ω(N). Note that Ω : B∆ → (0, 1] descends to a continuous map Ω̂ : B̂∆ →
(0, 1]. Suppose that for some x ∈ (δ − c, δ), Ω̂−1({x}) is countable. As dim(B̂∆) ≥ 2, B̂∆ \ Ω̂
−1({x})
is connected. Hence, Ω̂(B̂∆ \ Ω̂
−1({x})) is connected, a contradiction. In case that (δ, δ + r) ⊂ Ω(N) an
analogous argument concludes the proof.
Remark 3.3. The proof of Proposition 2.17 above also establishes the following: suppose that ∆ is an
n-dimensional Delzant polytope with at least n+ 3 facets and with Ω(∆) := δ ∈ (0, 1). Also suppose that
Ω(B∆) contains an open neighborhood (δ−c, δ+c) of δ. Then for each x ∈ (δ−c, δ+c), Ω
−1({x}) ⊂ Dn
is uncountable.
Proposition 2.15. Let Dn denote the set of equivalence classes of n-dimensional Delzant polytopes and
Ω : Dn → (0, 1]
be the maximal density function, n ≥ 2. Then Ω−1({x}) is uncountable for all x ∈ (0, 1).
Proof. By Remark 3.3, it suffices to show that there exists an n-dimensional Delzant polytope ∆ with at
least n + 3 facets for which Ω(B∆) = (0, 1). Consider the polytope ∆(ǫ1, ǫ2) obtained by removing from
the standard n-dimensional simplex an admissible simplex of radius ǫi at the vertex ei for i = 1, 2. For
compatible choices of ǫ1 and ǫ2, we obtain a Delzant polytope with n + 3 facets. Fix ǫ
1 and ǫ
2 both very
close to zero and let E ⊂ [0, 1]2 be the set of pairs (x, y) for which ∆(x, y) = ∆s for some s ∈ B∆(ǫ0
By definition, Ω({∆(x, y) | (x, y) ∈ E}) ⊂ Ω(B∆(ǫ01, ǫ02)), a connected subset of the interval (0, 1). We
conclude by showing Ω({∆(x, y)|(x, y) ∈ E}) contains deleted open neighborhoods of 0 and 1 in [0, 1].
To obtain a deleted neighborhood of 1 we remark that as (x, y) → (0, 0), (x, y) ∈ E and Ω(∆(x, y)) → 1.
Similarly, to obtain a deleted neighborhood of 0 we remark that as y → 0, the pairs of the form (1− 2y, y)
are in E and Ω(1− 2y, y) → 0.
We thank Professor Novik for bringing the example above to our attention. We conclude with the
following:
Question. Let [(M2n, σ, ψ)] be an equivalence class of symplectic-toric manifolds. Is there a formula
for the number of different maximal toric packings of M in terms of its equivariant symplectic invariants?
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Alvaro Pelayo
Department of Mathematics, University of Michigan
2074 East Hall, 530 Church Street, Ann Arbor, MI 48109–1043, USA
e-mail: apelayo@umich.edu
Benjamin Schmidt
Department of Mathematics, University of Chicago
5734 South University Avenue, Chicago, Illinois 60637
e-mail: schmidt@math.uchicago.edu
	Introduction and Main Result
	Motivation
	Main Results
	Reduction to Convex Geometry
	Proofs of propositions ??, ??, and ??.
