Background On Metric Geometry, I - Ohio State University
Background on Metric Geometry, IFacundo MémoliMath and CSE Departments, OSU.memoli@math.osu.edu2014
Table of Contents1. Background I: review of metric spaces‚ Compact metric spaces‚ Hausdorff distance‚ A question about nets‚ Maps between metric spaces‚ Some invariants of metric spaces‚ Approximation of metric spaces: a first attempt‚ Geodesic metric spaces‚ Different classes of metric spaces2. *
BibliographyBackground on MetricGeometry, IFacundo MémoliBackground I: review ofmetric spacesCompact metricspacesHausdorff distanceA question about netsMaps between metricspacesSome invariants ofmetric spacesApproximation ofmetric spaces: a firstattempt Burago, Burago and Ivanov’s book [BBI01]. Gromov’s book [Gro99]. do Carmo (Differential Geometry of Curves and Surfaces).Geodesic metricspacesDifferent classes ofmetric spaces*p. 3
Metric spaceBackground on MetricGeometry, IFacundo MémoliDefinition 1.1. A metric space is a pair pX, dX q where X is a set anddX : X ˆ X Ñ R with1Background I: review ofmetric spacesCompact metricspacesHausdorff distanceA question about nets1 dX px, x q “ 0 if and only if x “ x .11Maps between metricspaces1 dX px, x q “ dX px , xq for all x, x P X.Some invariants ofmetric spaces dX px, x2 q ď dX px, x1 q dX px1 , x2 q for all x, x1 , x2 P X.Approximation ofmetric spaces: a firstattemptGeodesic metricspacesOne says that dX is the metric or distance on X.If dX satisfies all but the first condition above, one says that dX is a semi-metricon X.Different classes ofmetric spaces*I will frequently refer to a metric space X with the implicit assumption that itsmetric is denoted by dX .Remark 1.1 (Distance to a set). Let S Ă X. We define the distance to S,dX p , Sq : X Ñ R byx ÞÑ inf dX px, sq.sPSp. 4
ExamplesBackground on MetricGeometry, IFacundo MémoliExample 1.1 (Restriction metric). Ω Ă Rn and dΩ pω, ω 1 q “ }ω ω 1 }.Background I: review ofmetric spacesCompact metricspacesExample 1.2 (Sn ). Spheres Sn with “intrinsic” metric. Consider Sn Ă Rn 1and for x, x1 P Sn this metric is given bydSn px, x1 q “ 2 arcsinHausdorff distanceA question about nets1}x x }2Example 1.3 (Ultrametrics). Finite set X and u : X ˆ X Ñ R which issymmetric and upx, x2 q ď max upx, x1 q, upx1 , x2 q for all x, x1 , x2 P X.Maps between metricspacesSome invariants ofmetric spacesApproximation ofmetric spaces: a firstattemptGeodesic metricspacesDifferent classes ofmetric 2x4r3r3r20x4r1r2r3Ultrametrics appear in many applications, including hierarchical clustering(dendrograms).p. 5
Compact metric spacesBackground on MetricGeometry, IFacundo MémoliBackground I: review ofmetric spacesCompact metricspacesHausdorff distanceA question about netsDefinition 1.2. Let X be a metric space and ε ą 0. A set S Ă X is calledan ε-net for X if dX px, Sq ď ε for all x P X.Maps between metricspacesSome invariants ofmetric spacesX is called totally bounded if for any ε ą 0 there is a finite ε-net for X.Approximation ofmetric spaces: a firstattemptDefinition 1.3. For a given ε ą 0 a set S in a metric space X is calledε-separated if dX ps, s1 q ě ε for all s, s1 P S.Excercise 1. Prove thatGeodesic metricspacesDifferent classes ofmetric spaces*1. if there exists a 3ε -net for X with cardinality n, then an ε-separated setin X cannot contain more than n points.2. A maximal separated ε-set is an ε-net.p. 6
Compact metric spacesBackground on MetricGeometry, IFacundo MémoliBackground I: review ofmetric spacesCompact metricspacesHausdorff distanceA question about netsAn openŤ covering of a topological space is any collection U of open sets suchthat U PU “ X.Maps between metricspacesSome invariants ofmetric spacesRecall that (by definition) a compact topological space X is one for which anyopen covering has a finite sub-collection that still covers X.Also, recall that a complete metric space is one for which Cauchy sequencesconverge. The sequence txn un Ă X is Cauchy, if for any ε ą 0 there existsN P N s.t. dX pxn , xm q ă ε for all n, m ą N .Approximation ofmetric spaces: a firstattemptGeodesic metricspacesDifferent classes ofmetric spaces*Theorem 1.1. Let X be a metric space. Then, X is compact if and only ifX is complete and totally bounded.We denote by G (for Gromov) the collection of all compact metric spaces.p. 7
Hausdorff distanceBackground on MetricGeometry, IFacundo MémoliBackground I: review ofmetric spacesLet X be a metric space. For a set S Ă X we denote by Bε pSq the set of allpoints x such that dX px, Sq ă ε.Compact metricspacesHausdorff distanceA question about netsDefinition 1.4 (Hausdorff distance). Let A, B P 2 . The Hausdorff distancebetween A and B is defined by dXH A, B :“ inftε ą 0 A Ă Bε pBq and B Ă Bε pAqu.XMaps between metricspacesSome invariants ofmetric spacesApproximation ofmetric spaces: a firstattemptExcercise 2. Prove thatˆ dXA,B“maxsupdpa,Bq,supdpb,Aq.XXHaPAbPBGeodesic metricspacesDifferent classes ofmetric spaces*Proposition 1.1. Let X be a metric space. Then X1. dXH , is a semi-metric on 2 . 2. dXH A, A “ 0 for any A Ă X. 3. If A, B Ă X are closed and dXH A, B “ 0, then A “ B.p. 8
Hausdorff distance, cont’dBackground on MetricGeometry, IFacundo MémoliBackground I: review ofmetric spacesCompact metricspacesHausdorff distanceLet CpXq denotethe collection of all closed subsets of X. Then, we have that pCpXq, dXH , q is a metric space. Furthermore, one has Theorem 1.2 (Blaschke). If pX, dX q is compact, then pCpXq, dXH , q is alsocompact.A question about netsMaps between metricspacesSome invariants ofmetric spacesApproximation ofmetric spaces: a firstattemptThat is, we have an application H : G Ñ G pX, dX q ÞÑ pCpXq, dXH , q.Geodesic metricspacesDifferent classes ofmetric spaces*Remark 1.2. For a given S Ă X, letCovRadX pSq :“ inftε ą 0 X Ď Bε pSqu. Clearly, CovRadX pSq “ dXH S, X .p. 9
Matlab code for Hausdorff distanceBackground on MetricGeometry, IFacundo MémoliBackground I: review ofmetric spacesCompact metricspacesHausdorff distanceA question about netsMaps between metricspacesSome invariants ofmetric spacesApproximation ofmetric spaces: a firstattemptGeodesic metricspacesDifferent classes ofmetric spacesTry coding the Hausdorff distance between finite subsets of the plane in Matlab.*p. 10
Background on MetricGeometry, IMinimal ε-nets in (compact) metric spacesFacundo MémoliBackground I: review ofmetric spacesCompact metricspaces1Lemma 1.1 (Marriage Lemma). Let Z ˇand Zˇ be finite sets with a relationK Ă Z ˆ Z 1 such that for any A Ă Z, ˇKpAqˇ ě A . Then, there exists abijection ϕ : Z Ñ Z 1 with pz, ϕpzqq P K for all z P Z.Hausdorff distanceA question about netsMaps between metricspacesProposition 1.2. Fix ε ą 0 and assume S, S 1 are two minimal ε-nets inX (assumed to be compact) with n “ npεq points each. Then, there exists abijection ϕ : S Ñ S 1 s.t. maxsPS dX ps, ϕpsqq ď 2ε.Some invariants ofmetric spacesLet R Ă S ˆ S 1 be given by all those ps, s1 q s.t. Bε psq X Bε ps1 q ‰ H. Pick anyA Ă S and note thatďRpAq :“ts1 P S 1 pa, sq P RuDifferent classes ofmetric spacesApproximation ofmetric spaces: a firstattemptGeodesic metricspacesProof.*aPA is s.t. RpAq ě A . Otherwise, consider N :“ RpAq Y SzA . Clearly, N is an ε-net forX and N ă npεq— this contradicts the fact that npεq is the minimal cardinality amongstall ε-nets of X. Then, apply the Marriage Lemma to conclude that there exists a bijectionϕ : S Ñ S 1 s.t. ps, ϕpsqq P R for all s P S. This means that Bε psq X Bε pϕpsqq ‰ H andhence dps, ϕpsqq ď 2ε.p. 11
Maps between metric spaces: DistortionBackground on MetricGeometry, IFacundo MémoliDefinition 1.5 (Distortion). Let pX, dX q and pY, dY q be metric spaces andf : X Ñ Y a map. The distortion of f is defined asˇˇ dis f :“ sup ˇdX px, x1 q dY pf pxq, f px1 qqˇ.Background I: review ofmetric spacesCompact metricspacesHausdorff distanceA question about netsx,x1 PXMaps between metricspacesSome invariants ofmetric spacesApproximation ofmetric spaces: a firstattemptGeodesic metricspacesDifferent classes ofmetric spaces* We say that a map f : X Ñ Y is distance preserving whenever dis f “ 0.An isometry between X and Y is any bijective map f : X Ñ Y which is inaddition distance preserving. One says that two metric spaces are isometric ifthere exists an isometry between them.The set IsopXq of all isometries f : X Ñ X is called the isometry group of X.p. 12
Isometry: the case of Euclidean setsBackground on MetricGeometry, IFacundo MémoliBackground I: review ofmetric spacesCompact metricspacesHausdorff distanceA question about netsMaps between metricspacesIf X and Y in the definition of isometry are both Rd , then we are looking atmaps T : Rd Ñ Rd such that }T ppq T pqq} “ }p q} for all points p, q P Rd .All such T arise as the composition of a translation and a orthogonal transformation. The set of all such maps is denoted Epdq and is called the EuclideanGroup. If we choose (canonical basis) coordinates on Rd , then any T P Epdqcan be represented as T ppq “ Q p b where Q is a d ˆ d orthogonal matrix (i.e.Q QT “ I) and b is (translation) vector.Some invariants ofmetric spacesApproximation ofmetric spaces: a firstattemptGeodesic metricspacesDifferent classes ofmetric spaces*If Q is an orthogonal matrix, then }Q p} “ }p} for all points p. This means thatwhenever T P Epdq, then ă T ppq, T pqq ą“ă p, q ą for all points p, q.p. 13
Isometry: the case of finite Euclidean setsBackground on MetricGeometry, IFacundo MémoliBackground I: review ofmetric spacesCompact metricspacesLemma 1.2 (Folklore Lemma). Let A “ ta1 , a2 , . . . , an u and B “tb1 , b2 , . . . , bn u two sets in Rd such that }ai aj } “ }bi bj } for alli, j “ 1, . . . , n. Then, there exists an isometry of ambient space T : Rd Ñ Rdsuch that bi “ T pai q for each i.Hausdorff distanceA question about netsMaps between metricspacesSome invariants ofmetric spacesThis is actually quite interesting: from purely intrinsic information (the interpointdistances of points in A and B we can deduce the existence of an extrinsic object(the transformation T ) with strong properties. This happens because we areassuming that both A and B are “special”, namely they live in Euclidean space.Approximation ofmetric spaces: a firstattemptGeodesic metricspacesDifferent classes ofmetric spaces*Excercise 3. Try to prove this theorem for youself. Or try to find and studythe proof.p. 14
Isometric embeddingBackground on MetricGeometry, IFacundo MémoliBackground I: review ofmetric spacesCompact metricspacesHausdorff distanceA question about netsDefinition 1.6. A map f : X Ñ Y between metric spaces X and Y is anisometric embedding of X into Y if f pXq Ă Y endowed with the restrictionof the metric from Y is isometric to X.Maps between metricspacesSome invariants ofmetric spacesApproximation ofmetric spaces: a firstattemptExample 1.4. S with the intrinsic metric does not admit an isometricembedding into any Rk , k P N.2Excerciseembedding). Let X be a compact metric space. 4 (Kuratowski’s Consider CpXq, } }L8 , the metric space of all real valued continuous functions on X, where the metric is the L8 norm. Attach to each x P X thefunction fx :“ dX p , xq : X Ñ R . Then,}fx fx1 }8 “ dX px, x1 qGeodesic metricspacesDifferent classes ofmetric spaces*for all x, x1 P X.p. 15
Lipschitz maps and dilatationBackground on MetricGeometry, IFacundo MémoliBackground I: review ofmetric spacesCompact metricspacesHausdorff distanceA question about netsMaps between metricspacesSome invariants ofmetric spacesDefinition 1.7. A map f : X Ñ Y between metric spaces X and Y is calledLipschitz if there exists L ě 0 s.t.dY pf pxq, f px1 qq ď L dX px, x1 qApproximation ofmetric spaces: a firstattemptGeodesic metricspacesfor all x, x1 P X.Any such L is called a Lipschitz constant for f . The minimal Lipschitzconstant of a map f is called the dilatation of f and denoted dil f .Different classes ofmetric spaces*A map with Lipschitz constant 1 is called non-expanding.p. 16
Distance preserving maps in compact spacesBackground on MetricGeometry, IFacundo MémoliBackground I: review ofmetric spacesCompact metricspacesHausdorff distanceA question about netsMaps between metricspacesA compact metric space cannot be isometric to a proper subset of itself.Some invariants ofmetric spacesTheorem 1.3. Let X be a compact metric space and f : X Ñ X be adistance preserving map. Then f pXq “ X.Approximation ofmetric spaces: a firstattemptGeodesic metricspacesProof.Assume p P Xzf pXq. Since f pXq is compact (and hence closed) there exists ε ą 0s.t. Bε ppq X f pXq “ H. Let n be the maximal cardinality of an ε separated set in X andS Ă X be an ε-separated set with cardinality n. Then, f pSq is also ε-separated. Also,Different classes ofmetric spaces*dX pp, f pSqq ě dX pp, f pXqq ě εand therefore f pSq Y tpu is also ε-separated but with cardinality n 1, which contradicts themaximality of n.p. 17
Non-expanding maps in compact spacesBackground on MetricGeometry, IFacundo MémoliTheorem 1.4. Let X be a compact metric space. Then,Background I: review ofmetric spacesCompact metricspacesHausdorff distance1. Any non-expanding surjective map is an isometry.A question about nets2. If a map f : X Ñ X is non-contracting: dX pf pxq, f px1 qq ě dX px, x1 qfor all x, x1 P X, then f is an isometry.Maps between metricspacesSome invariants ofmetric spacesApproximation ofmetric spaces: a firstattemptProof.We prove (1). Assume p, q are such that dX pf ppq, f pqqq ă dX pp, qq for somep, q P X. Fix such a pair of points and pick ε ą 0 s.t. dX pf ppq, f pqqq ă dX pp, qq 5ε. Let nbe such that there exists at least one ε-net in X of cardinality n. Let Nn Ă X n the collectionof all n-tuples of points that form ε-nets in X. This set is closed in X and hence compact.Define the function D : X n Ñ R given bypx1 , . . . , xn q ÞÑnÿi,j“1Geodesic metricspacesDifferent classes ofmetric spaces*dX pxi , xj q.This function is continuous and therefore attains a minimum on Nn . Let S “ px1 , . . . , xn q PNn be a minimizer. Since f is non-expanding and surjective then f pSq P Nn . Also, Dpf pSqq ďDpSq and hence dX pxi , xj q “ dX pf pxi q, f pxj qq for all i, j P t1, . . . , nu.Let i0 and j0 be s.t. dX pp, xi0 q, dX pq, xj0 q ď ε. Then, one has dX pxi0 , xj0 q ě dX pp, qq 2εand dX pf pxi0 q, f pxj0 qq ď dX pf ppq, f pqqq 2ε ď dX pp, qq 3ε. This givesdX pxi0 , xj0 q ą dX pf pxi0 q, f pxj0 qq, a contradiction.p. 18
Diameter, inradius, etceteraBackground on MetricGeometry, IFacundo MémoliLet X P G. DefineBackground I: review ofmetric spacesCompact metricspaces separation: X ÞÑ sep pXq :“ mintdX px, x1 q, x ‰ x1 u.Hausdorff distanceA question about nets1 diameter: X ÞÑ diam pXq :“ maxx,x1 dX px, x q.Maps between metricspaces1 inradius: X ÞÑ rad pXq :“ minx maxx1 dX px, x q. eccentricity: X ÞÑ eccX : X Ñ R . It is given by eccX pxq “maxx1 dX px, x1 q, x P X. kˆk curvature sets: pX, kq ÞÑ Kk pXq Ă R , the collection of all k ˆ ksymmetric matrices ppdX pxi , xj qqq where px1 , x2 , . . . , xk q P X k .Some invariants ofmetric spacesApproximation ofmetric spaces: a firstattemptGeodesic metricspacesDifferent classes ofmetric spaces* Packing and covering numbers and others: consider things like 1řk– xtk pXq :“ C2kmaxt iąj dX pxi , xj q, px1 , . . . , xk q P X k u,– covk pXq :“ mintε ě 0 s.t. exists ε-net S for X, with S “ ku– capk pXq :“ maxtε ě 0 s.t. exists S with S “ k and sep pSq ě εuQuestion 1.1. What happens to these invariants if I “perturb” X slightly?How can one define a notion of perturbation of metric spaces?p. 19
some more invariantsBackground on MetricGeometry, IFacundo MémoliBackground I: review ofmetric spacesCompact metricspacesHausdorff distanceA question about netsMaps between metricspacesSome invariants ofmetric spacesApproximation ofmetric spaces: a firstattempt ( CapNbrpX, εq :“ max S ; S Ă X with Bε{2 pxi q X Bε{2 pxj q “ H, i ‰ j CovNbrpX, εq :“ min t S ; S Ă X with X Ď Bε pSquExcercise 5. Prove that CapNbrpX, εq ě CovNbrpX, εq. for all ε ě 0.Geodesic metricspacesDifferent classes ofmetric spaces*p. 20
ε-IsometriesBackground on MetricGeometry, IFacundo MémoliBackground I: review ofmetric spacesCompact metricspacesHausdorff distanceWe seek a relaxation of the notion of isometries. We need to preserve distancesalright, but we also need to make sure we fill in the target space with the imageof the source via the “approximate isometry map”. This suggests:A question about netsDefinition 1.8. One says that a map f : X Ñ Y is a ε-isometry betweenmetric spaces X and Y ifApproximation ofmetric spaces: a firstattempt dis f ď ε and f pXq is an ε-net for Y .Maps between metricspacesSome invariants ofmetric spacesGeodesic metricspacesDifferent classes ofmetric spaces*Note that we do not require f to be continuous.Excercise 6. Prove that a 0-isometry coincides with an isometry in theusual sense.p. 21
Length structuresBackground on MetricGeometry, IFacundo MémoliBackground I: review ofmetric spacesCompact metricspacesConsider a metric space pX, dX q. We’ll construct a new metric d X over X. Foreach continuous path γ : r0, 1s Ñ X we consider its lengthLdX pγq :“ suptNÿi“1Hausdorff distanceA question about netsMaps between metricspacesdX pγpti q, γpti 1 qquSome invariants ofmetric spacesApproximation ofmetric spaces: a firstattemptwhere the supremum is taken over all partitions of r0, 1s. A set of points P “tt1 , t2 , . . . , tN u is a partition of r0, 1s if 0 “ t1 ď t2 ď ď tN “ 1. We saythat a curve γ is rectifiable whenver its length is finite.The intrinsic metric d X on X is defined as follows: for each pair x, x1 P X weconsider ΓX px, x1 q the set of all continuos paths joining x to x1 . Then, we defineGeodesic metricspacesDifferent classes ofmetric spaces*d X px, x1 q :“ inftLdX pγq; γ P ΓX px, x1 qu.Notice that if the points x and x1 cannot be connected by a continous curve,then the definition above does not make sense. So when that is the case, onewould (informally) say that d X px, x1 q “ 8. Notice that this can be the caseeven if dX px, x1 q is finite.p. 22
Geodesic metric spacesBackground on MetricGeometry, IFacundo MémoliConsider a metric space pX, dX q. We’ll construct a new metric d X over X. Foreach continuous path γ : r0, 1s Ñ X we consider its lengthLdX pγq :“ suptNÿi“1Background I: review ofmetric spacesCompact metricspacesHausdorff distancedX pγpti q, γpti 1 qquA question about netsMaps between metricspacesSome invariants ofmetric spaceswhere the supremum is taken over all partitions of r0, 1s. A set of points P “tt1 , t2 , . . . , tN u is a partition of r0, 1s if 0 “ t1 ď t2 ď ď tN “ 1. We saythat a curve γ is rectifiable whenver its length is finite.d XGeodesic metricspaces1The intrinsic metricon X is defined as follows: for each pair x, x P X weconsider ΓX px, x1 q the set of all continuos paths joining x to x1 . Then, we defined X px, x1 qApproximation ofmetric spaces: a firstattemptDifferent classes ofmetric spaces*1:“ inftLdX pγq; γ P ΓX px, x qu.Notice that if the points x and x1 cannot be connected by a continous curve,then the definition above does not make sense. So when that is the case, onewould (informally) say that d X px, x1 q “ 8. Notice that this can be the caseeven if dX px, x1 q is finite.If d X “ dX , then one says that dX is intrinsic and that pX, dX q is a path metricspace or also sometimes length space. One says that pX, dX q is geodesic if forany pair of points x, x1 there exists γ P ΓX px, x1 q such that LdX pγq “ dX px, x1 q.p. 23
A zoo of metric spacesBackground on MetricGeometry, IFacundo MémoliBackground I: review ofmetric spacesCompact metricspacesHausdorff distanceA question about nets We saw examples of different metrics on spheres.Maps between metricspacesSome invariants ofmetric spaces We saw ultrametric spaces.Approximation ofmetric spaces: a firstattempt More general are trees.Geodesic metricspaces Trees
p. 6 Compact metric spaces Definition 1.2. Let X be a metric space and ε ą 0. A set S Ă X is called an ε-net for X if d Xpx,Sqďε for all x P X. X is called totally bounded if for any ε ą 0thereisafiniteε-net for X. Definition 1.3. For a given ε ą 0asetS in a metric sp
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