The goal of this section is to prove the existence of a minimizer for the Shape distance-problem. The basic idea is that the problem should be reduced to having a continuous map on a compact subset of the isometry group and the minimizer of this map should be exactly the minimizer of the shape distance-problem. This is why we start with the following lemma and its important corollary.
Lemma 6.21. LetGbe a topological group with continuous group action onX. Then the group action ofG onWp(X) is continuous as well.
Proof. Let ((gi),(µi))i∈I be a net in G×Wp(X) with (gi, µi) → (g, µ) (see Appendix A.1.1). As we know from Section 2.1, the Wasserstein distance metrizes weak convergence, i.e. we have to show that Wp(giµi, gµ) → 0 as (gi, µi)→(g, µ). For this we see that
Wp(giµi, gµ) =Wp(µi, gi−1gµ)≤Wp(µi, µ) +Wp(µ, gi−1gµ) ∀i∈I.
We know already thatµi* µ. But it is also true thatgi−1gµ * µ: Sincegi→g in Gand Gacts continuously onX, the functionsg−ig(x) converge pointwise againstIdG(x). Using dominated convergence theorem yieldsgi−1gµ * µand thus the continuity of the Wasserstein distance onWp(X).
Remark 6.22. In caseG⊂ISO(X), there is always at least one topology onG such that the action on X is continuous, namely the compact-open topology (see A.1.1 for a definition). Generally, the following holds: IsX a locally compact andY any topological space and is furtherH ⊂C(X, Y), whereC(X, Y) is the set of continuous functionsf :X→Y. Then one can show ([Sch69], p.74) that the compact-open topology is the coarsest topology on H such that the map H×X→Y, (Φ, x)7→Φ(x) is continuous.
Corollary 6.23. Let Gand (X, d)be as in Lemma 6.21. Then the map G→ R;g7→Wp(gµ, ν)is continuous for every choice ofµ, ν∈Wp(X).
Proof. We know from Theorem 2.28 that Wp(·,·) is continuous on Wp(X)× Wp(X). With this, Wp(·, ν) is continuous on Wp(X) for every choice of ν.
Since we know from Lemma6.21 that the group action ofGonWp(X) is also continuous, we have continuity as a composition of continuous functions.
The map considered in Corollary 6.23 is of course the map for which we want to find the existence of a minimizer. The complement of the compact set on which this map is supposed to be considered for this should consist of all those elements inGwhich transports the two given measures “too far away”, so that the distance between them is increasing instead of decreasing. For this, it is useful to consider spaces on which most of the mass of probability measures is concentrated on compact balls, to have better control over their Wasserstein distances. To synthesize this, the action should “take care” of compact sets appropriately. The property we will use for this, similarly to Chapter 3, is proper.
Definition 6.24 (Proper action). Let X be a metric space. A continuous actionG×X →X of a group G onX is called proper, if the map G×X → X×X, (g, x)7→(gx, x) is proper.
For a defintion of proper maps, see Definition3.2.
Remark 6.25. The definition for an action to be proper is equivalent to saying that for everyK1, K2⊂X compact the set{g∈G|gK1∩K26=∅}is compact inG.
One can show that ifGacts proper onX thenX/Gis a Hausdorff space.
Definition 6.26 (Proper metric space). A metric space is calledproper if every bounded closed set is compact.
Remark 6.27. • A metric space (X, d) being proper is equivalent to x 7→
d(x0, x) being a proper map for anyx0∈X.
• A proper metric space is automatically locally compact (which is why Wp(X) cannot be proper in general). This fact enables us to use, the for our situation suitable, Definition6.24of proper actions on locally compact spaces.
• Any proper space is complete.
• One can view “proper” as a finiteness condition, recalling that finite di-mensional Hilbert spaces are always proper, but in infinite dimensions one can show that the closed unit ball is not sequentially compact.
Now we have all the ingredients we need to formulate the theorem.
Theorem 6.28 (Existence of a minimizer). Let the Polish metric space (X, d)be proper. Additionally let the topology and the action ofGbe such that it acts properly onX. In this case a minimizer for the probleminfg∈GWp(gµ, ν) exists for every choice ofµ, ν∈ Pp(X).
Corollary 6.29. In the setting of Theorem 6.28the Shape distance Dp(·,·) is a metric.
Proof of Corollary6.29. Since we already know thatDp(·,·) is a pseudometric, we only have to make sure that if Dp(µ, ν) = 0 then automatically [µ] = [ν].
From Theorem6.28we know that there exists an element gmin ∈Gsuch that 0 =Dp(µ, ν) =Wp(gminµ, ν). Since Wp(·,·) is a metric we can conclude that gminµ=ν and thus [µ] = [ν].
Remark 6.30. If there exists no minimizer of infg∈GWp(gµ, ν) in general, one can always convert the pseudometric Dp(·,·) into a metric via the so called metric identification. For this procedure one identifies points which have zero pseudodistance with each other.
We will use the following definition in the proof for the existence of a mini-mizer.
Definition 6.31.LetXbe a topological space. A sequence of subsets (Kn)n∈N⊂ X is called exhaustion of X ifS
n∈NKn=X andKn⊂Kn+1.
Proof of Theorem6.28. IfX is bounded, then it is automatically compact and then alsoGis compact. So from now on, let X be unbounded.
According to the prerequisites, for everyR∈R, x∈X: BR(x) is compact and we have an exhaustion ofX by balls centered around an arbitrary element x∈X. Now letµ, ν ∈ Pp(X) and x∈X. Using the Monotone Convergence Theorem, for every >0 there exists an element R∈Rsuch thatµ(BR(x))>
1− andν(BR(x))>1−. Then letC :=Wp(µ, ν)p and chooseg ∈Gsuch that gBR0(x)∩BR0(x) = ∅, whereR0 = kR, k > 1
2R
p
r C
1−2 + 1. To fulfill this, we have to choose <1/2.
If no suchgexists, we know automatically that Gis compact and the proof ends here. Otherwise, forB0:=BR0(x), it is
Wp(gµ, ν)p
= inf
Π∈Adm(gµ,ν)
Z
gB×B
dp(x, y)dΠ(x, y) + Z
X×X\gB×B
dp(x, y)dΠ(x, y)
!
≥ (2(R0−R))p inf
Π∈Adm(gµ,ν)Π(gB×B)≥(2(k−1)R)p(1−2)
> C
This means that the minimizer has to be found in the subset {g ∈ G | gBR0(x)∩BR0(x)6=∅} 6=∅. Since all closed balls of (X, d) are compact and we
assumed the group action to be proper, this subset is compact, so the existence of a minimizer is guaranteed.
Remark 6.32. One might have the idea that, instead of demanding that there exists an exhaustion ofX by compact balls, it is enough to say that there exists a compact exhaustion, i.e. to say thatX is σ-compact. But in this case, it is not clear how to control the distances of elements of the setgKn0 to elements of the setKn0, whereKn are the elements of the exhaustion.
To confirm intuition we additionally give the following proposition, which is implicitly used in the preceding proof:
Proposition 6.33. Let µ∈ Pp(X) and g ∈G. Then supp(gµ) = g supp(µ).
In particular, two measures cannot be equivalent to each other if each support is not an isometric image of the respective other support.
Proof. Let g ∈ G, µ ∈ Pp(X). To establish this proposition, we need the following two identities:
1.) {A∈ B(X)|µ(g−1(Ac)) = 0}={gA∈ B(X)|µ(Ac) = 0}.
2.) LetI be an index family, we then have \
i∈I
gAi=g\
i∈I
Ai forAi⊂X.
Thereby isgA:={ga|a∈A}, forA⊂X. Proof of the first identity:
LetA∈ B(X), thenA=gg−1A, sincegis bijective. Then we haveµ((g−1A)c) = µ(g−1Ac) = 0. For the other way round, let gA∈ B(X), thenµ(g−1(gA)c) = µ(Ac) = 0. We will use the second identity without further justification. Now we can see that
suppgµ = \
gµ(Ac)=0
A= \
µ(g−1Ac)=0
A= \
µ(Ac)=0
gA
= g \
µ(Ac)=0
A=g \
µ(Ac)=0
A=g suppµ