A maximum principle for pointwise energies of quadratic Wasserstein minimal networks

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by  Jonathan Dahl

A MAXIMUM PRINCIPLE FOR POINTWISE ENERGIES OF
QUADRATIC WASSERSTEIN MINIMAL NETWORKS
1
1 JONATHANDAHL
0
2
Abstract. We show that suitable convex energy functionals on a quadratic

n Wasserstein space satisfy a maximum principle on minimal networks. We
a explore consequences of this maximum principlefor the structure of minimal
J networks.

5
2

]
P
A 1. Introduction
.
h Given k points p ,...,p in a geodesic space1Y, one canask for a minimal net-
1 k
at workspanning p1,...,pk. For a complete, connected Riemannian manifold M, the
m space of Borel probability measure P(M) may be metrized, allowing infinite dis-
tances,bytheWassersteindistanceW associatedtotheoptimaltransportproblem
[ 2
for quadratic cost c = d2. Furthermore, as shown by McCann [9], the component

3
P (M) containing the point masses is a geodesic space. Continuing the investiga-
v 2
tions in [3], we examine the structure of such W -minimal networks by considering
6 2
3 gradient flows of suitable convex energy functionals on P2(M). In particular, we
2 establish conditions under which the energy is maximized at a boundary measure
0 of a minimal network. The book of Ambrosio, Gigli, and Savar´e [2] is an excellent
1. sourceforboththehistoryandgeneralresultsofgradientflowsinP2(Rn);Villani’s
1 book [15] serves a parallel role for optimal transport problems.
0 For M =Rn, we will use the fact that eachminimal networkin P (Rn) is also a
2
1
solutionofan associatedmulti-marginaloptimaltransportationproblem. A recent
:
v result of Pass [11] reduces uniqueness of solutions of multi-marginal problems to
Xi computations in terms of the cost function. We will perform these computations
assuming that the underlying graph of the minimal network is a star and at least
r
a one boundary point is absolutely continuous, and use this uniqueness to show that
any non-expansive map fixing the boundary points must fix the network.
We beginbydefining Wassersteinspace,the multi-marginaloptimaltransporta-
tion problems, and the minimal network problems. In Section 3, we examine the
specialcaseofdifferentialentropyinEuclideanspace. Wethengeneralizeourmax-
imum principle to an abstract class of energy functionals in Section 4. Finally, in
Section5,westrengthenourmaximumprincipleinthespecialcaseofminimalstars
in P (Rn) by examining the corresponding multi-marginal optimal transportation
2
problem.
Theauthor wassupportedbytheNSFunderRTGgrantDMS-0838703.
1Ageodesicspaceisametricspacewithanintrinsicmetricsuchthatanytwopointsarejoined
byashortestpath.
1
2 JONATHANDAHL
2. Background
First, we introduce the multi-marginal problem, for comparison with our qua-
dratic Wasserstein minimal network problem defined at the end of the section.
Given a collection of spaces X ,...,X and a collection of subsets of proba-
1 m
bility measures P(X ),..., P(Y ), we define the set of transport
1 1 m m
P ⊂ P ⊂
plans Π( ,..., ) as the set of probability measures π P(X X )
1 m 1 m
P P ∈ × ···×
such that (proj ) π for all i. The multi-marginal Kantorovich problem
for an infinitesiXmial#cos∈t fPuinction c : X X R and boundary data
1 m
× ··· × →
µ P(X ),...,µ P(X ) is then to minimize
1 1 m m
∈ ∈
C(π):= c(x ,...,x )dπ
1 m
ZX1×···×Xm
over all transport plans π Π(µ ,...,µ ). A minimizing transport plan is called
1 m
∈
an optimal transport plan. This is a natural relaxation of a corresponding multi-
marginal Monge problem, in which one must minimize
C(T ,...,T ):= c(x ,T (x ),...,T (x ))dµ
2 m 1 2 1 m 1 1
ZX1
over all (m 1)-tuples of measurable maps T : X X such that (T ) µ = µ
i 1 i i # 1 i
− →
for all i. For each (T ,...,T ), the corresponding transport plan (T ,...,T ) µ
2 m 2 m # 1
is called deterministic.
In the case m = 2, we recover the standard Kantorvich and Monge problems.
WeareparticularlyinterestedinthecaseX =X =X withc(x ,x )=d2(x ,x )
1 2 1 2 1 2
for a metric d on X. Taking the square root of the minimal cost, we define
1/2
W (µ ,µ )= inf d2(x ,x )dπ .
2 1 2 1 2
(cid:18)π∈Π(µ1,µ2)ZX2 (cid:19)
Assuming (X,d) is complete, separable, locally compact geodesic space, W is a
2
metric on
P (X):= µ P(X): d2(x,x )dµ< ,
2 0
∈ ∞
(cid:26) ZX (cid:27)
calledthe Wassersteindistance of order2,and (P (X),W ) is a geodesic space([9]
2 2
for the Riemannian case, Corollary 7.22 of [15] for the general case).2
If c(x ,...,x ) = c (x ,x ), the cost C(π) of a transport plan π can be
1 m i<j ij i j
interpretedas the totalcostofthe networkoftransportplans (proj proj ) π
P Xi× Xj #
with respect to the infinitesimal costs c . If X = = X = X, for some fixed
ij 1 m
cˆ : X2 R each c is either cˆ or identically 0, an··d·the cˆ-Kantorovich problem
ij
→
happenstodefineametricdˆonP(X)where(P(X),dˆ)isageodesicspace,thenfor
a c-optimal transport plan π, C(π) is precisely the dˆ-length of the corresponding
networkin(P(X),dˆ). Ifwethenremovetheconstraintonthelastm k marginals
−
of π, the problem of minimizing
C(π)= c(x ,...,x )dπ
1 m
ZXm
over all transport plans
π Π(µ ,...,µ ,P(X),...,P(X))
1 k
∈
2Herex0∈X isanarbitrarypoint.
A MAXIMUM PRINCIPLE FOR POINTWISE ENERGIES 3
isequivalentto the problemoffindingalengthminimizing networkin(P(X),dˆ)in
the class of networks parameterized by a specific graph G of m vertices v ,...,v
1 m
with v ,...,v mapping to µ ,...,µ respectively. Here, two vertices v ,v in G,
1 k 1 k i j
i<j, are adjacent if and only if c =cˆ.
ij
The formulation just mentioned will not work for the case where (P (X),W )
2 2
is a geodesic space, so we consider now the standard minimal network problems
for a metric space. We define a network in a metric space Y as a continuous
map Γ : G Y, where G is a connected graph metrized to have edges of length
→
1. All graphs G in what follows are connected unless stated otherwise. Given k
vertices in a graph G and k points p ,...,p Y, the G-parameterized minimal
1 k
∈
networkproblemistofindanetworkΓwithimageofminimallengthsubjecttothe
constraintsΓ(v )=p ,...,Γ(v )=p . Wedefinealsothegeneralminimalnetwork
1 1 k k
problem for p ,...,p : find a network Γ:G Y of minimal length subject to the
1 k
→
constraint p ,...,p Γ(G). A solution Γ is called a minimal network. Note that
1 k
∈
G is allowed to vary in the general problem. Our main focus will be on minimal
network problems for Y =(P (X),W ), a quadratic Wasserstein space.
2 2
3. Euclidean differential entropy
Under certain geometric assumptions on the base space X, regularity of the
vertices of a solution of the general minimal network problem in (P (X),W ) can
2 2
lead to combinatorial regularity of the solution. In particular,
Theorem 1 ([3]). Suppose M is a Riemannian manifold with isometric split-
ting M = S Rn where S is compact with nonnegative sectional curvature, and
×
µ ,...,µ P (M) have compact support. Then there is a minimal network in
1 k 2
∈
P (M) spanning µ ,...,µ . Furthermore, this solution has a canonical representa-
2 1 k
tive Γ:G P (M) such that
2
→
(1) Γ is injective.
(2) G is a tree.
(3) Vertices in G not mapped to µ ,...,µ have degree at least three.
1 k
(4) For any vertex v in G Γ−1( µ ,...,µ ), if Γ(v) is absolutely continuous
1 k
\ { }
with respect to the volume measure, then the degree of v is three and all
pairs of geodesics in Γ(G) meeting at Γ(v) do so with an angle of 2π/3.
This result follows from the inner product structure of the tangent cone at ν
∈
Pac(M) demonstrated by Lott and Villani [8] for ν of compact support. In the
2
specialcaseM =Rn,wemaythereforeremovethecompactsupporthypothesis[2].
Itisnaturaltoaskthenifabsolutecontinuity(w.r.t. Lebesguemeasure n)ofeach
L
measureµ intheboundarydataimpliesabsolutecontinuityfortheSteinersolution,
i
i.e. Γ(G) Pac(Rn). In this direction, we recall the definition of Shannon’s [13]
differential⊂ent2ropy h:P (Rn) [ ,+ ),
2
→ −∞ ∞
ρ(x)logρ(x)d n(x) if µ=ρ n,
h(µ)= − Rn L L
(−∞R otherwise.
Proposition 2. If µ ,...,µ Pac(Rn) are taken as boundary data for a graph G
1 k ∈ 2
andm=min h(µ )> ,thenthereisasolutionoftheG-parameterizedminimal
i i
network problem contai−n∞ed entirely in Pac(Rn).
2
Proof.
(µ)=max( m, h(µ))
F − −
4 JONATHANDAHL
defines a convex functional :P (Rn) ( ,+ ] with effective domain
2
F → −∞ ∞
D( ):= −1(R) Pac(Rn).
F F ⊂ 2
As in Theorem 11.2.1 of [2], we obtain a non-expansive gradient flow for on
F
D( )=P (Rn).3
2
TFhe flow is instantly regularizing, so the flowed network lies in Pac(Rn) for all
2
t > 0. Furthermore, is minimized and constant on each µ in the boundary
i
F
data, so they remain fixed under the flow. Therefore, if we start with an injective
solution Γ : G P (Rn) of the G-parameterized minimal network problem [3],
2
non-expansivene→ss of the flow provides a solution Γ :G Pac(Rn) for t>0. (cid:3)
t → 2
Recall that the Steiner ratio of a metric space X is
L (M)
s
inf ,
M La(M)
where M is any finite set of points in X, L (M) is the infimum of the lengths of
s
allnetworksspanning M, andL (M) is the lengthofthe minimal spanning tree of
a
M.
Corollary 3. The Steiner ratio of P (Rn) is at least 1/√3.
2
Proof. P (Rn) is an Alexandrov space on nonnegative curvature (see [8] or [14]
2
for stronger statements). Propostion 2, Theorem 1, and Toponogov’s theorem
for Alexandrov spaces allow us to apply the arguments of Graham and Hwang
[6] to solutions of the general minimal network problem, provided the boundary
data µ ,...µ Pac(Rn) have finite differential entropy. Approximating arbitrary
1 k ∈ 2
boundary data by finite linear combinations of characteristic functions of balls in
Rn, we may bound the Steiner ratio. (cid:3)
4. Maximum principle for admissible energies
In this section, we will introduce a suitable generalizationof(negative) differen-
tial entropy for which a maximum principle on minimal networks holds. This will
allow us to obtain greater control of the structure of minimal networks.
Definition 4. Aproper,lowersemi-continuousfunctionalφ:P (X) ( ,+ ]
2
→ −∞ ∞
for a space X is called admissible if:
(1) D(φ) = φ−1(R) is a geodesic space, and φ is convex along geodesics in
D(φ).
(2) For each a [ ,+ ), t > 0, there is a non-expansive map ψ :
a,t
∈ −∞ ∞
P (X) D(φ), called the a-stopped flow of φ at t.
2
→
(3) The set of fixed points for ψ is φ−1( ,a].
a,t
−∞
We are thinking of ψ as the gradient flow for φ stopped at the value a. The
a,t
usualconvexityconditionforsuchfunctionalsisweakdisplacementconvexity,where
convexityis only assumedalongsomegeodesic betweeneverytwo points inP (X).
2
It may therefore be more natural to consider the convexity condition as weak dis-
placement convexity combined with uniqueness of geodesics in D(φ), even though
the condition in our definition is strictly weaker.
More importantly, we require the flow to exist on all of P (X), and hence want
2
eitherD(φ)=P (X)orD(φ)=P (X)withanappropriateregularizationestimate
2 2
3Thisissimplythegradientflowfor−h,orheatflow,stoppedatm.
A MAXIMUM PRINCIPLE FOR POINTWISE ENERGIES 5
inordertoobtainψ (D(φ)) D(φ). Existenceofsuchregularizinggradientflows
a,t
⊂
has been announced by Savar´e [12], under mild geometric assumptions on X. In
particular, X can be any complete Riemannian manifold, or Alexandrov space,
with curvature bounded below and finite diameter. Similar results were obtained
independently by Ohta for any proper, lower semi-continuous convex functional
φ:P (X) ( ,+ ] for a compact Alexandrov space X [10], although here we
2
→ −∞ ∞
donotnecessarilyhaveψ (D(φ)) D(φ). Intheparticularcaseofrelativeentropy
a,t
⊂
w.r.t. volume measure, admissibility of φ is shown by Erbar [4] for any connected,
complete Riemannian manifold X with a lower bound on Ricci curvature.
We now present our maximum principle.
Theorem 5. Suppose X is a complete, separable, locally compact, non-branching
geodesic space and let φ : X ( ,+ ] be an admissible functional. For
→ −∞ ∞
µ ,...µ P (X) and m = max φ(µ ), any solution Γ : G P (X) of the
1 k 2 i i 2
∈ →
G-parameterized minimal network problem for boundary data µ ,...,µ has
1 k
Γ(G) φ−1( ,m].
⊂ −∞
Proof. If m= , there is nothing to prove, so assume m< . Let Γ =ψ Γ.
t m,t
∞ ∞ ◦
ψ is non-expansive and µ ,...,µ are fixed by ψ , so Γ also solves the G-
m,t 1 k m,t t
parameterized minimal network problem. In particular, the image of each edge is
a geodesic, and ψ is continuous on Γ (G) by convexity. Fix t > 0. Let M =
t 0
max φ, and suppose that M > m. G is a finite graph, so we may choose
Γt0(G)
a (m,M) such that φ=a on vertices of Γ (G).
∈ 6 t0
Consider ψ Γ . For some edge e of Γ , ψ a on an initial interval e′
a,s ◦ t0 1 t0 ≤ 1
of e while ψ > a at the opposite endpoint v . As before, ψ (e ) is a geodesic,
1 2 a,s 1
but ψ (e′) = e′ while ψ (v ) = v . We therefore have a branched geodesic in
a,s 1 1 a,s 2 6 2
P (X), contradicting the fact that X is non-branching by Corollary 7.32 of [15].
2
Thus M =m. (cid:3)
Corollary 6. Suppose M is a Riemannian manifold with isometric splitting M =
S Rn where S is compact with nonnegative sectional curvature, µ ,...,µ
1 k
× ∈
P (M) have compact support, and max φ(µ ) < + for an admissible functional
2 i i
∞
φ. Then there is a minimal network in D(φ) spanning µ ,...,µ . Furthermore, if
1 k
D(φ) Pac(M), this solution has a canonical representative Γ:G P (M) such
⊂ 2 → 2
that
(1) Γ is injective.
(2) G is a tree.
(3) Vertices in G not mapped to µ ,...,µ have degree three.
1 k
(4) For any vertex v in G Γ−1( µ ,...,µ ), all pairs of geodesics in Γ(G)
1 k
\ { }
meeting at Γ(v) do so with an angle of 2π/3.
Again, for M = Rn, compactness of supports is unnecessary. Typical candi-
datesfor φwith D(φ) Pac(Rn)areinternalenergies,suchas differentialentropy,
⊂ 2
relative entropy with respect to an absolutely continuous, log-concave reference
measure, and the power functional
(ρ(x))md n(x) if µ=ρ n,
(µ)= Rn L L
F (+R ∞ otherwise,
for some fixed m>1.
6 JONATHANDAHL
Applying our work again to the special case of finite linear combinations of
characteristic functions of balls, we may generalize Corollary 3.
Corollary 7. If M is a Riemannian manifold with isometric splitting M =S Rn
×
where S is compact with nonnegative sectional curvature, then the Steiner ratio of
P (M) is at least 1/√3.
2
Proof. We may clearly reduce to the compact case by projection in M, where
relative entropy
ρ(x)logρ(x)dγ if µ=ργ,
(µ)= M
F (R+∞ otherwise,
for the volume measure γ defines an admissible functional as in [4] or [12]. The
result now follows as before for Corollary 3. (cid:3)
5. A related multi-marginal problem
We now motivate a related problem in the uniqueness theory of multi-marginal
problems. Let l 3, and consider the G-parameterized minimal network problem
≥
of minimizing
l
Φ(ν):= W (µ ,ν)
2 i
i=1
X
forsomefixedboundarydataµ ,...,µ P (Rn),whereGisthestarwithlleaves.
1 l 2
∈
Suppose ν minimizes Φ, and assume we are in the nontrivial case
0
W (µ ,ν )>0, 1 i l.
2 i 0
≤ ≤
For
σ :=1/W (ν ,µ ),
i 2 0 i
ν is also a minimizer of
0
l
Ψ(µ):= σ W2(µ ,µ),
i 2 i
i=1
X
as the corresponding l-plane and ellipsoid are tangent.
l
Ψ(µ)= σ inf x x 2dπ
i 1 2
i=1 π∈Π(µi,µ)ZR2n| − |
X
l
= inf σ x x 2dπ,
i i j
π∈Π(µ1,...,µl,µ)ZR(l+1)n i=1 | − |
X
so
l
inf Ψ(ν)= inf σ x x 2dπ.
i i l+1
ν∈P2(Rn) π∈Π(µ1,...,µl,P2(Rn))ZR(l+1)n i=1 | − |
X
For simplicity, suppose that µ ,...,µ P (Rn) have compact support, take
1 l 2
∈
R large enough that the support of each µ is contained in B (0), and set B =
i R
B (0).
R+1
A MAXIMUM PRINCIPLE FOR POINTWISE ENERGIES 7
By standard duality arguments, the infimum is achieved and clearly supported
in B, with
l l
min σ x x 2dπ = sup φ dµ φ ,
i i l+1 i i l+1 ∞
π∈Π(µ1,...,µl,P2(B))ZBl+1 i=1 | − | (φ1,...,φl+1)∈S i=1ZB !−k k
X X
for
l+1 l
S = (φ ,...,φ ) (C (B))l+1 : x ,...,x B, φ (x ) σ x x 2 .
1 l+k b 1 l+1 i i i i l+1
( ∈ ∀ ∈ ≤ | − | )
i=1 i=1
X X
If we let
l l
x¯:= σ x σ
i i i
i=1 !(cid:30) i=1 !
X X
and
l l
S′ = (φ ,...,φ ) (C (B))l : x ,...,x B, φ (x ) σ x x¯2 ,
1 l b 1 l i i i i
( ∈ ∀ ∈ ≤ | − | )
i=1 i=1
X X
wemaysuitablyadjustany(φ˜ ,...,φ˜ ) S toobtainanelementofS′ bytaking
1 l+1
∈
φ =φ˜ +M,φ =φ˜ ,...,φ =φ˜
1 1 2 2 l l
for
M =min 0, minφ˜ (x) .
l+1
x∈B
(cid:18) (cid:19)
(φ ,...,φ ,0) S by construction and 0 =0, so
1 l ∞
∈ k k
l l
sup φ dµ φ = sup φ dµ .
i i l+1 ∞ i i
(φ1,...,φl+1)∈S i=1ZB !−k k (φ1,...,φl)∈S′ i=1ZB !
X X
The right hand side is also dual for the multi-marginal Kantorovich problem with
infinitesimal cost c(x ,...,x )= l σ x x¯2, so
1 l i=1 i| i− |
l P l
min σ x x 2dπ = min σ x x¯2dπ.
i i l+1 i i
π∈Π(µ1,...,µl,P2(B))ZBl+1 i=1 | − | π∈Π(µ1,...,µl)ZBl i=1 | − |
X X
We may also see this directly by noting that the map f(x ,...,x)=x¯ satisfies
1 l
l l
σ x x 2d(π f π)= σ x x¯2dπ
i i l+1 # i i
| − | × | − |
ZBl+1 i=1 ZBl i=1
X X
forall π Π(µ ,...,µ ), andanyminimizer inΠ(µ ,...,µ ,P (B)) musthavethis
1 l 1 l 2
∈
form by the minimizing property of f.4 In particular, uniqueness of the minimizer
ν of Ψ would follow from uniqueness of a minimizer for the multi-marginal Kan-
0
torovichproblem for µ ,...,µ and c(x ,...,x )= l σ x x¯2. Furthermore,
1 l 1 l i=1 i| i− |
such uniqueness would imply that ν =ν is the unique solution of the system
0
P
W (µ ,ν)=W (µ ,ν ),...,W (µ ,ν)=W (µ ,ν ).
2 1 2 1 0 2 l 2 l 0
4In fact, the equality continues to hold for B = Rn, arbitrary µ1,...,µl ∈ P2(Rn), and
arbitrarygraphsG,withtheminimumontheleftachievedbyourassumedsolutionoftheminimal
networkproblem,andtheminimumontherightachievedbyasolutionoftheKantorovichproblem
[7].
8 JONATHANDAHL
This would give a result similar to our maximum principle above, without refer-
encing the potential φ of the non-expansive flow. In fact, any non-expansive map
fixing the boundary data µ ,...,µ would be forced to fix ν as well.
1 l 0
Proposition 8. The multi-marginal Kantorovich problem for µ ,...,µ P (Rn)
1 l 2
with compact support, µ Pac(Rn), and ∈
1 ∈ 2
l
c(x ,...,x )= σ x x¯2
1 l i i
| − |
i=1
X
has a unique solution for any collection of positive weights σ . Hence, any non-
i
expansive map F :P (Rn) P (Rn) with F(µ )=µ ,...,F(µ )=µ must fix the
2 2 1 1 l l
→
free vertex ν of a solution of the G-parameterized minimal network problem, where
0
G is the star whose l leaves are the fixed vertices.
Proof. Forl =2,theresultissimplygeodesicuniquenessandfollowsfromBrenier’s
theorem. Assume l 3.
The following con≥ditions for infinitesimal cost functions c:M M R
1 m
×···× →
on precompact5 Riemannian manifolds M ,...,M of dimension n were used by
1 m
Pass [11] to obtain uniqueness of solutions to multi-marginal problems:
(1) c C2(M M ).
1 m
∈ ×···×
(2) c is (1,m)-twisted, meaning the map x D c(x ,...,x ) from M to
m 7→ x1 1 m m
T∗ M is injective for all fixed x , k =m.
x1 1 k 6
(3) cis(1,m)-non-degenerate,meaningD2 c(x ,...,x ):T M T∗ M
x1xm 1 m xm m → x1 1
is injective for all (x ,...,x ).
1 m
(4) For all choices of y = (y ,...,y ) M M and of y(i) =
1 m 1 m
∈ × ··· ×
(y (i),...,y (m)) M M suchthaty (i)=y fori=2,...,m 1,
1 i 1 m i i
∈ ×···× −
we have
T <0,
y,y(2),y(3),...,y(m−1)
for T defined below.
y,y(2),y(3),...,y(m−1)
Theorem 9 ([11]). If M ,...,M and c satisfy the above conditions and µ
1 m 1
∈
P(M ) does not charge sets of Hausdorff dimension less than or equal to n 1,
1
−
then the multi-marginal Kantorovich and Monge problems have unique solutions
for any µ P(M ),...,µ P(M ).
2 2 m m
∈ ∈
We now define T :
y,y(2),y(3),...,y(m−1)
m−1 m−1
S := D2 c(y)
y − xixj
j=2 i=2,i6=j
X X
m−1
+ (D2 c(D2 c)−1D2 c)(y),
xixm x1xm x1xj
i,j=2
X
m−1
H := (Hess c(y(i)) Hess c(y)),
y,y(2),y(3),...,y(m−1) xi − xi
i=1
X
T :=S +H .
y,y(2),y(3),...,y(m−1) y y,y(2),y(3),...,y(m−1)
5Precisely,eachMi issmoothlyembeddedinsomemanifoldNi,inwhichMi iscompact.
A MAXIMUM PRINCIPLE FOR POINTWISE ENERGIES 9
Figure 1. A P (Rn) minimal network with boundary data
2
µ ,...,µ and lengths a,b as labeled.
1 4
µ µ
1 3
µ ..........................................................a.......................................................................................................................................................a..............................................................ν................1 b ν2.........................................................................................................................................................a............................................................a................................................................................µ4
2
For our infinitesimal cost c(x ,...,x ) = l σ x x¯2, we may compute
directly in natural coordinates an1d find tlhat at (ip=1,..i.|,ip−) |Rln,
1 l
P ∈
n σ2 2σ σ
D c(p ,...,p )= 2 σ i pα i j pα dxα,
xi 1 l α=1 i− lk=1σk! i − j6=i lk=1σk j i
X X
 
P P
and
σ2
2 σ i dxα dxα if i=j,
Dxixjc(p1,...,pl)=Pαα − l2σii−σσjPdxlkαi=1⊗σdkx!αj i ⊗ i if i6=j.
k=1 k
Therefore, c is C2, (1,l)-twistePd, anPd (1,l)-non-degenerate. H 0, so
≡
l−1 2σ2
T = − i dxα dxα <0. (cid:3)
l σ i ⊗ i
i=2 k=1 k α
X X
P
The condition µ Pac(Rd) cannot be removed, as seen by considering
1 ∈ 2
1 1 1 1
µ = δ + δ , µ = δ + δ ,
1 2 (1,0) 2 (−1,0) 2 2 (0,1) 2 (0,−1)
and the isometry F on P (Rn) induced by reflection about the x-axis. Only one of
2
the geodesics from µ to µ is fixed by F.
1 2
Most of the preceding discussion may be generalized to an arbitrary graph G,
where x¯ is replaced by the collection of points in Rn solving the appropriately
weightedminimization problemfor the sumofsquareddistances. However,we can
have T < 0 in this case, causing the end of the proof of Proposition 8 to fail. In
6
particular, if we take a minimal network as in Figure 1, computations similar to
our previous work show that T <0 if and only if a >√2b. Switching the labeling
of µ and µ only makes the inequality worse, with T <0 if and only if a>4b.
2 4
10 JONATHANDAHL
Appendix A. L∞ regularity and a stronger maximum principle
Shortly after the initial posting of this paper, the author was made aware of a
recent preprint of Agueh and Carlier [1] in which the problem of minimizing
l
Ψ(µ)= σ W2(µ ,µ)
i 2 i
i=1
X
overµ P (Rn)is considered. Inthis appendix,wediscussbrieflytheapplications
2
∈
of their work to minimal network problems. Specifically, we obtain L∞ estimates
and derive a stronger maximum principle for minimal networks in P (Rn).
2
Building on the work of Gangbo and S`wi¸ech [5], Agueh and Carlier obtain the
following L∞ regularity of solutions.
Theorem 10 (Theorem 5.1 and Remark 5.2 of [1]). Let µ ,...,µ P (Rn) and
1 l 2
∈
let σ ,...,σ be positive reals summing to 1/2. Assume µ L∞, i.e. µ = ρ n
1 l 1 1 1
∈ L
with ρ L∞. If Ψ(ν)=min Ψ(µ), then
1 µ
∈
1
kνkL∞ ≤ σnkµ1kL∞.
1
We may always rescale our coefficients so σ = 1/2, thereby obtaining such
i i
anestimatelocally(onastar)foraminimalnetwork. Movinginwardfromµ along
i
stars, we obtain an interior L∞ estimate. P
Corollary 11. Let Γ : G P (Rn) be a G-parameterized minimal network span-
2
ning µ ,...,µ P (Rn).→If G has k edges and the length of the image of each
1 l 2
∈
edge is in [1/M,1/m], then for any ν Γ(G) with distance at least 1/M from a
∈
boundary vertex,
kn
2kM
ν L∞ µ1 L∞.
k k ≤ m k k
(cid:18) (cid:19)
For a global L∞ estimate, we may consider the above estimate away from the
boundary combined with further applications of Theorem 10 on each leaf.
Corollary 12. Let Γ : G P (Rn) be a G-parameterized minimal network span-
2
ning µ ,...,µ P (Rn). I→f G has k edges and the length of the image of each edge
1 l 2
∈
is in [1/M,1/m], then for any ν Γ(G) and any λ>1,
∈
n kn
λ 2kM
ν L∞ max , max µi L∞ .
k k ≤" ((cid:18)λ−1(cid:19) (cid:18) λm (cid:19) )#(cid:18)i=1,...,lk k (cid:19)
In these estimates, we get controlof 1/m and k essentially for free, but we need
some prior knowledge of a lower bound for 1/M. Therefore, for a given set of
boundary data, global L∞ estimates are reduced to a qualitative estimate on the
non-degeneracyofedges. Notealsothattheinteriorestimatemaygiveinformation
in cases where the global is trivial due to µi L∞ = for some i.
k k ∞
Agueh and Carlier also define a notion of convexity along barycentersof a func-
tional :P (Rn) R bytakingaweightedaverageofthevaluesof forcompar-
2
F → F
isonwith (ν) forthe barycenter,or minimizer of Ψ, ν. They showthat convexity
F
along barycenters follows from convexity along generalized geodesics, and hence
holds for the standard admissible functionals discussed above. Again, a simple in-
duction along stars allows us to obtain a bound for on a minimal network by a
F
weighted average of the values of on the boundary data, yielding a maximum
F