Lecture 3 - The Gromov-Hausdorff Topology
February 4, 2010
Gromov-Hausdorff distance and the Gromov-Hausdorff topology are central to these
lectures.
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Equivalent formulations of the Gromov-Hausdorff distance
Proposition 1.1 The Gromov-Hausdorff distance dGH (X, Y ) is the
` infimum of the Hausdorff distances between X and Y taken among all metrics on X Y that restrict to the
given metric on X and on Y .
Pf
`
Define d˜GH = inf{dH (X, Y )}, where the infimum is taken over metrics on Z = X Y
that restrict to the given metrics on X and Y . Since dGH is an infimum taken over a larger
set,
dGH ≤ d˜GH .
Now consider a metric on some ambient space Z`that restricts
to the given metrics on X and
`
Y . Put α = dH (X, Y ). Define a function d¯ : X Y ×X Y → R≥0 as follows. If x1 , x2 ∈ X
¯ 1 , x1 ) = d(x1 , x2 ), d(y
¯ 1 , y2 ) = d(y1 , y2 ), d(x
˜ 1 , y1 ) = d(x1 , y1 ) if
and y1 , ye ∈ Y , set d(x
¯
d(x1 , y1 ) ≥ α/2, and d(x1 , y1 ) = α/2 if d(x1 , y1 ) < α/2. We check that the triangle
inequality holds. By the symmetry of the distance function we only have to check that
˜ y) ≤ d(x,
˜ x0 ) + d(x
˜ 0 , y).
d(x,
There are four cases. First if d(x, y) ≥ α/2 and d(x0 , y) ≥ α/2 then
˜ y) = d(x, y) ≤ d(x, x0 ) + d(x0 , y) = d(x,
˜ x0 ) + d(x
˜ 0 , y).
d(x,
Second if d(x, y) ≥ α/2 and d(x0 , y) < α/2 then
˜ y) = d(x, y) ≤ d(x, x0 ) + d(x0 , y) < d(x,
˜ x0 ) + α/2 < d(x,
˜ x0 ) + d(x
˜ 0 , y).
d(x,
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Third if d(x, y) < α/2 and d(x0 , y) ≥ α/2 then
˜ y) = α/2 ≤ d(x0 , y) < d(x, x0 ) + d(x0 , y) = d(x,
˜ x0 ) + d(x
˜ 0 , y).
d(x,
Finally if d(x, y) < α/2 and d(x0 , y) < α/2 then
˜ y) = α/2 = d(x
˜ 0 , y) < d(x,
˜ x0 ) + d(x
˜ 0 , y).
d(x,
`
Therefore from isometric embeddings X ,→ Z, y ,→ Z, we have found a metric on X Y ,
that restricts to the given metrics on X and Y , and so that the Hausdorff distance from X
to Y is preserved. This proves that
d¯GH (X, Y ) ≤ dGH .
A map f : X → Y (not necessarily continuous) between metric spaces is called an
-GHA (for “Gromov-Hausdorff approximation”) if |dY (f (x1 ), f (x2 )) − dX (x1 , x2 )| < for
all x1 , x2 ∈ X, and Y is in the -neighborhood of f (X). We can define a new distance
function between metric spaces, called dd
GH , by setting
0
dd
GH (X, Y ) = inf{ > 0 | there are −GHA s f : X → Y and g : Y → X }.
It is a simple exercise to prove that this is a metric: if there is an 1 -GHA f : X → Y and
an 2 -GHA g : Y → Z, then the composition satisfies
|dZ (gf (x1 ), gf (x2 )) − dX (x1 , x2 )|
≤ |dZ (gf (x1 ), gf (x2 )) − dY (f (x1 ), f (x2 ))| + |dY (f (x1 ), f (x2 )) − dX (x1 , x2 )|
≤ 1 + 2
and it is also easy to show that the (1 + 2 )-neighborhood of f g(X) is Z. Taking infima,
d
d
we have that dd
GH (X, Z) ≤ dGH (X, Y ) + dGH (Y, Z).
Proposition 1.2 The metrics dd
GH and dGH are equivalent (though they are not the same).
Pf
We prove that any sequence that converges in one metric converges in the other. If
Xi → X in the dGH sense, we can easily construct
` 2-approximations fi : Xi → X. To
do this, note that for all big enough i, since Xi X has a metric in which X is in the
-neighborhood of Xi (and vice-versa), we can pick any map that sends a point p ∈ Xi to
some point f (p) ∈ X a distance at most 2 away from p. This can be done, for instance, by
choosing a denumerated, finite set of points XF ⊂ X that is -dense (by the compactness
of X), and sending any point p ∈ Xi to the nearest point of XF . If two or more points are
equally close to p, then send p to the point of XF that is lower in the denumeration. Then
for p, q ∈ Xi we have |d(p, q) − d(f (p), f (q))| < 2, and X is clearly in the 2-neighborhood
of f (Xi ).
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`
Conversely, if Xi → X in the dd
X, for
GH -topology, we can construct metrics on Xi
large enough i in which X is in the 2-neighborhood of Xi . Construct a distance function
d so that if fi : Xi → X is a 2-approximation put d(xi , f (xi )) = , and given any other
xi ∈ Xi and x ∈ X,
d(xi , x) = + 0inf (d(xi , x0i ) + d(f (x0i ), x)) .
xi ∈Xi
We verify the triangle inequality. First assume xi , x0i ∈ Xi . The only case to verify is
d(xi , x0i ) ≤ d(xi , x) + d(x, x0i ), where x ∈ X. We have
d(xi , x) + d(x, x0i )
inf (d(xi , x00i ) + d(f (x00i ), x)) +
=
2 +
≥
2 +
≥
2 + d(xi , x0i ) ≥ d(xi , x0i ).
x00
i ∈Xi
inf (d(x0i , x00i ) + d(f (x00i ), x))
x00
i ∈Xi
inf (d(xi , x00i ) + d(f (x00i ), x) + d(x0i , x00i ) + d(f (x00i ), x))
x00
i ∈Xi
Next assume xi ∈ Xi and x ∈ X. If x0 ∈ X then
d(xi , x)
inf (d(xi , x00i ) + d(f (x00i ), x))
=
+
≤
+
=
d(xi , x0 ) + d(x0 , x),
=
+
≤
+
=
d(xi , x0i ) + d(x0i , x).
x00
i ∈Xi
inf (d(xi , x00i ) + d(f (x00i ), x0 ) + d(x0 , x))
x00
i ∈Xi
and if x0i ∈ Xi then
d(xi , x)
inf (d(xi , x00i ) + d(f (x00i ), x))
x00
i ∈Xi
inf (d(xi , x0i ) + d(x0i , x00i ) + d(f (x0i ), x))
x00
i ∈Xi
Finally assume x, x0 ∈ X. Given any xi ∈ Xi then
d(x, x0 ) < 2 + infx00i ∈Xi (d(f (x00i ), x) + d(f (x00i ), x0 ))
≤ 2 + infx00i ∈Xi (d(xi , x00i ) + d(f (x00i ), x) + d(xi , x00i ) + d(f (x00i ), x0 ))
≤ 2 + infx00i ∈Xi (d(xi , x00i ) + d(f (x00i ), x)) + infx00i ∈Xi (d(xi , x00i ) + d(f (x00i ), x0 ))
= d(x, xi ) + d(xi , x0 )
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Properties of the Gromov-Hausdorff metric
Proposition 2.1 dGH is a metric on the set of compact metric spaces, modulo isometry.
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Pf
If X and Y are isometric then clearly dGH (X, Y ) = 0.
Conversely
assume dGH (X, Y ) = 0. Then there is a sequence of distance functions di
`
on X Y with di |X = dX and di |Y = dY so that di,H (X, Y ) → 0. Let j > 0 be a sequence
that converges to 0. For each j construct finite sets of points Xj = {xk } and Yj = {yk }
with the following properties: Xj is j -dense in X, Yj is j -dense in Y , and for large enough
i the sets Xj and Yj are j -close
S in the Hausdorff metric. We
S also require that Xj ⊂ Xj+1
and Yj ⊂ Yj+1 , so that X = j Xj is dense in X and Y = j Yj is dense in Y .
Now consider the distance functions {di } restricted to Xj ∪ Xj . Because this set is
finite, a subsequence dij converges to a limiting pseudometric dj . Passing to ever more
refined subsequences of di as j increases and taking a diagonal subsequence (which
` we also
call di ), we get convergence`
to a pseudometric d on X ∪ Y, a dense subset of X Y , and
therefore convergence on X Y .
Given any j , a given point x ∈ X is j -close to a point xj ∈ X , which is j -close to
a point of yj ∈ Y. Taking a limit y = limj yj we have that d(x, y) = 0. Similarly given an
arbitrary point y ∈ Y we can find a point x ∈ X with d(x, y) = 0.
`
Identify points a, b ∈ X Y with d(a, b) = 0 and call the moduli space M . Since
d|X = dX and d|Y = dY and the triangle inequality holds, a point in X is identified with
a unique point in Y , and vice-versa. We now have natural isometric equivalences X → M
and Y → M so that X and Y are isometric.
Proposition 2.2 The Gromov-Hausdorff topology on the set of compact metric spaces is
second countable.
Pf
If a topology is Hausdorff and separable it is second countable, or better, a separable
metric space is second countable. Consider the set X̃ of finite metric spaces where all
distances are rational. There are countably many such spaces. To see that these spaces are
dense, consider a compact metric space X. We can construct a sequence Xi of such finite
spaces that converge to X by letting Xi ⊂ X be a 2−i -dense set of points. The metric space
Xi has a distance of less than 2−i from some finite metric space X̃i with rational distance,
so that X̃i → X. This proves that X̃ is dense.
As it happens, the pointed Gromov-Hausdorff distance is not second countable. We
have already constructed an uncountable collection of subsets of R1 that are Hausdorff
distance 1 from each other.
Lemma 2.3 (Gromov’s Precompactness Lemma) Let N : N → N be monotonic. Assume M is a collection of metric spaces so that each M ∈ M has a 1j -dense discrete subset
of cardinality ≤ N (j). Then M is precompact.
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Pf
Let {Mi } ⊂ M, and let M̃i,j ⊂ Mi be a 1j -dense subset of cardinality ≤ N (j). By replacing
Pj
N (j) with i=1 N (i) we can assume that M̃i,j ⊂ M̃i,j+l . Fixing j and letting i → ∞ we
get convergence of M̃i,j along a subsequence to a space M̃j . Passing to further refinements
of the subsequence and taking a diagonal sequence, we get a sequence of distance functions
S
dk that converge on each M̃j , and therefore on M̃ = j M̃j . Now given > 0 there is an i
so that M̃i is -close to M̃ , and there is a j so that Mi,j is -close to both M̃i and to Mi .
Thus Mi converges to M̃ .
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