A remark on two notions of flatness for sets in the Euclidean space
aa r X i v : . [ m a t h . M G ] F e b A remark on two notions of flatness for sets in the Euclideanspace
Ivan Yuri Violo ∗ February 26, 2021
Abstract
In this note we compare two ways of measuring the n -dimensional “flatness” of a set S ⊂ R d , where n ∈ N and d > n . The first one is to consider the classical Reifenberg-flatnumbers α ( x, r ) ( x ∈ S , r > B r ( x ) between S and n -dimensional affine subspaces of R d . The secondis an ‘intrinsic’ approach in which we view the same set S as a metric space (endowedwith the induced Euclidean distance). Then we consider numbers a ( x, r )’s, that are thescaling-invariant Gromov-Hausdorff distances between balls centered at x of radius r in S and the n -dimensional Euclidean ball of the same radius.As main result of our analysis we make rigorous a phenomenon, first noted by Davidand Toro, for which the numbers a ( x, r )’s behaves as the square of the numbers α ( x, r )’s.Moreover we show how this result finds application in extending the Cheeger-Coldingintrinsic-Reifenberg theorem to the biLipschitz case.As a by-product of our arguments, we deduce analogous results also for the Jones’numbers β ’s (i.e. the one-sided version of the numbers α ’s). Contents a ≤ α and b ≤ β
84 Converse inequalities : α ≤ a , β ≤ b In this short note we consider two ways of measuring the “flatness” of a set in the Euclideanspace. The first one is by considering its best approximation by affine planes: more precisely,given a set S ⊂ R d and n ∈ N , with n < d , one defines α ( x, r ) := r − inf Γ d H ( S ∩ B r ( x ) , Γ ∩ B r ( x )) , for every r > x ∈ S, (1.1)where d H is the Hausdorff distance and where the infimum is taken among all the n -dimensional affine planes Γ containing x . ∗ SISSA, [email protected]
Gromov-Hausdorff distance , inparticular given S ⊂ R d and n ∈ N , with n < d , we set a ( x, r ) := r − d GH ( B ( S, d Eucl ) r ( x ) , B R n r (0)) , for every r > x ∈ S, where ( S, d Eucl ) is the metric space obtained by endowing S with the Euclidean distance.It follows immediately from the definition of d GH that a ( x, r ) ≤ α ( x, r ) , for every r > x ∈ S . (1.2)Moreover it is easy to build many examples for which α ( x, r ) ≤ a ( x, r ) , with a ( x, r ) = 0 and arbitrary small (one can take S to be a segment with a very smallinterval removed from its center). This shows that in general (1.2) cannot be improved andleads to the intuition the the quantities α ( x, r ) and a ( x, r ) are in some sense equivalent.However there are non-trivial cases in which the stronger inequality a ( x, r ) ≤ α ( x, r ) holds. The key example is the one of a very thin triangle: let P, Q ∈ R be the two pointsin the upper half plane that are at distance 1 from the origin and at distance ε ∈ (0 , / x -axis and let S ⊂ R be the union of the two closed segments joining P and Q tothe origin O . It can be immediately seen that α ( O, ≥ ε , while projecting S orthogonallyonto the x -axis easily shows that a ( O, r ) ≤ ε .The aim of this note is to explore the above phenomenon, that is to clarify to whichextent and in which cases the quantities a ( x, r ) behave like the square of the quantities α ( x, r ) . To state our main result we need the following notation: fixed ε ∈ (0 , / S ⊂ R d and n ∈ N with n < d , for every i ∈ Z we set α i := sup x ∈ S ∩ B (0) α ( x, − i ) , a i := sup x ∈ S ∩ B − ε (0) a ( x, − i ) , (1.3)where we neglected the dependence on ε , n and S . Our main result reads as follows: Theorem A.
For every n ∈ N there exists δ ( n ) > such that the following holds. Let S ⊂ R d , with d > n , ε ∈ (0 , / and define the numbers α i , a i as in (1.3) . Suppose that α i ≤ δ for every i ≥ ¯ i − , for some ¯ i ∈ N with − ¯ i < ε , then X i ≥ ¯ i a λi < C λ X i ≥ ¯ i − α λi , ∀ λ > , (1.4) where C λ is a positive constant depending only on λ and n . In particular for every λ > itholds that X i ≥ α λi < + ∞ = ⇒ X i ≥ a λi < + ∞ . (1.5)Theorem A will follow from a ‘weak’ version of the inequality a ( x, r ) . α ( x, r ) (seeTheorem 3.1), which as said above cannot hold in its ‘strong’ form.It has to be said that the fact that the numbers a ( x, r )’s “behaves” as the square ofnumbers α ( x, r )’s was already noted, at least at an informal level, by David and Toro (see[8]). However, to the author’s best knowledge, both the statement and the proof of TheoremA are new. 2 he Jones’ numbers β We will also prove the analogue of Theorem A for the “one sided”-version of the numbers α ( x, r ): given a set S ⊂ R d and n ∈ N , with n < d , we set β ( r, x ) := r − inf Γ sup y ∈ S ∩ B r ( x ) d ( y, Γ) , for every r > x ∈ S, (1.6)where the infimum is taken among all the n -dimensional affine planes Γ containing x . Thenumbers β ( x, r ) are usually refer to as L ∞ - Jones’ numbers (see for example [9], [2] and[11]). It is immediate from the definition that β ( x, r ) ≤ α ( x, r ) , for every x ∈ S and r > . We then define the following metric analogue of β ( x, r ): for a set S ⊂ R d and n ∈ N ,with n < d , we set b ( r, x ) := r − inf n δ : there exists a δ -isometry f : ( S ∩ B R d r ( x ) , d Eucl ) → ( B R n r (0) , d Eucl ) o , for every r > x ∈ S (see Section 3 for the definition of δ -isometry). As for thenumbers α ( x, r ) and a ( x, r ) we have the immediate inequality b ( x, r ) ≤ β ( x, r ) , for every r > x ∈ S . (1.7)Similarly to the numbers α i , a i we define for a given S ⊂ R d , n ∈ N with n < d , a fixed ε ∈ (0 , /
2) and every i ∈ Z β i := sup x ∈ S ∩ B (0) β ( x, − i ) , b i := sup x ∈ S ∩ B − ε (0) b ( x, − i ) , (1.8)where we neglected the dependence on ε , n and S . Then we can prove the following: Theorem B.
For every n ∈ N there exists δ ( n ) > such that the following holds. Let S ⊂ R d , with d > n, ε ∈ (0 , / and define the numbers β i , b i as in (1.8) . Suppose that α i ≤ δ for every i ≥ ¯ i − , for some ¯ i ∈ N with − ¯ i < ε (where α i are as in (1.3) ), then X i ≥ ¯ i b λi < C λ X i ≥ ¯ i − β λi , ∀ λ > , (1.9) where C λ is a positive constant depending only on λ and n . In particular Theorem B implies that, whenever lim sup i → + ∞ α i < δ (with δ as in thestatement of the theorem), we have that X i ≥ β λi < + ∞ = ⇒ X i ≥ b λi < + ∞ , (1.10)for every λ > b ( x, r ) . β ( x, r ) , which is also contained in Theorem 3.1. Converse inequalites
3n the last section we will prove that, contrary to the inequality ‘ a ( x, r ) . α ( x, r )’, theopposite estimate α ( x, r ) . a ( x, r ) holds in full generality. Moreover we will also prove that β ( x, r ) ≤ b ( x, r ) holds, provided B r ( x ) ∩ S contains n ‘sufficiently’ independent points. Thiscombined with the results in Theorem A and B shows that the intrinsic ‘Gromov-Hausdorff’approach and the ‘extrinsic’ Hausdorff approach to measuring ‘flatness’ in the Euclideanspace are in a sense equivalent up to a square factor. Motivations and application to Reifenberg’s theorem
We now explain the role, consequences and motivations of the results in this note.It is essential to recall that the quantities α ( x, r ) and β ( x, r ) coupled with smallness orsummability conditions as in (1.5), (1.10) (also called Dini-conditions) are tightly linked toparametrization and rectifiability results for sets in the Euclidean space. The more classicalare the celebrated Reifenberg theorem ([15]) and the rectifiability results of Jones ([11]),but there are also more recent and sophisticated works containing variants, generalizationsand refinements of these type of statements (see for example [16], [8], [9] and the referencestherein). It also worth to mention the the works in [14], [10] and [1], which contain similarresults, but where L p -versions of the β -Jones’ numbers are considered.There has been recently a growing interest in extending statements for sets in R d asabove (or rectifiability results in general), to the setting of metric spaces. The most notableinstance of this is the intrinsic-Reifenberg theorem by Cheeger and Colding ([5]), which hasrecently found many applications especially in the theory of singular metric spaces withsynthetic curvature conditions (see for example [5], [13] and [12]).If one is interested in extending to the setting of metric spaces results in R d involving thequantities α ( x, r ) and β ( x, r ) (or variants of them), it is more convenient to consider insteadthe numbers a ( x, r ) and b ( x, r ). This is because the the quantities α ( x, r ) , β ( x, r ) are con-fined to the Euclidean space, while their “Gromov-Hausdorff” counterparts are immediatelygeneralized to the metric setting. For this reason it is essential to have a good understandingof the relation between the numbers α ( x, r ) , β ( x, r ) and the numbers a ( x, r ) , b ( x, r ). Thisis the point in which Theorem A and Theorem B find their relevance.Indeed our Theorem A shows that it makes sense to interpret the numbers α ( x, r ) asthe square of a ( x, r ), at least when one is interested in their decay behaviour. To explainfurther the consequence of this fact we now show that Theorem A is crucial if one wants toextend the biLipschitz version of Reifenberg theorem to the metric setting. Let us also saythat this problem is what originated the writing of this note.We first need to recall the classical Reifenberg’s theorem: Theorem 1.1 ([15]) . For every n, d ∈ N with n < d there exists δ = δ ( n, d ) such that thefollowing holds. Let S ⊂ R d be closed, containing the origin and such that α i < δ, ∀ i ∈ N , where α i are as in (1.3) . Then there exists a biH¨older homeomorphism F : Ω → S ∩ B R n / (0) ,where Ω is an open set in R n . It was proven by Toro that if we require, besides smallness, also a fast decay of thenumber α i as i → + ∞ , the biH¨older regularity of the map F can be improved to biLipschitz.In particular we have the following: 4 heorem 1.2 ([16]) . For every ε > , n, d ∈ N with n < d , there exists δ = δ ( n, d, ε ) > such that the following holds. Let S ⊂ R d be closed, containing the origin and such that X i ≥ α i < δ, (1.11) where α i are as in (1.3) . Then there exists a (1+ ε )-biLipschitz homeomorphism F : Ω → S ∩ B R n / (0) , where Ω is an open set in R n . It is a remarkable result by Cheeger and Colding that Reifenberg’s theorem can begeneralized to metric spaces:
Theorem 1.3 ([5]) . For every n ∈ N there exists ε = ε ( n ) > such that the following holds.Let ( Z, d ) be a complete metric space, let z ∈ Z and define a i := sup z ∈ B / ( z ) i d GH ( B − i ( z ) , B R n − i (0)) , i ∈ N . Suppose that a i ≤ ε, ∀ i ∈ N . Then there exists a biH¨older homeomorphism F : Ω → B / ( z ) , where Ω is an open set in R n . As said above this result found recently a wide range of applications, in particular inthe study of regularity of singular metric spaces. It is therefore natural to ask weatheralso an analogous of Theorem 1.2 holds in the metric setting. A careful analysis of thearguments in[5] shows that, with little modifications, they can be adapted to prove thefollowing biLipschitz version of Theorem 1.3:
Theorem 1.4.
For every n ∈ N and ε > there exists δ = δ ( n, ε ) > such that thefollowing holds. Under the notations and assumptions of Theorem 1.3 suppose that X i ≥ a i < δ. (1.12) Then there exists a (1+ ε )-biLipschitz homeomorphism F : Ω → B / ( z ) , where Ω is anopen set in R n . Comparing the above with Theorem 1.2, the presence of the summability assumption(which is stronger than square summability) might lead to think that something strongerthan Theorem 1.4 should hold, at least for ‘nicer’ metric spaces, like subsets of the Euclideanspace. Indeed if one restricts its attention only to (1.2) (which as we said, cannot beimproved), the summability assumption (1.12) for a subset S of the Euclidean space (whenregarded as metric space ( S, d Eucl )) appears stronger than (1.11). Therefore it may seemthat Theorem 1.2 is in a sense missing some of the informations contained in the theoremof Toro.However the key observation is that Theorem A implies thatTheorem 1.4 is stronger than Theorem 1.2 , (1.13)where by “stronger” we mean that any set S satisfying the hypotheses of Theorem 1.2 alsosatisfies the hypotheses of Thoerem 1.4 (when regarded as metric space ( S, d Eucl ))).Finally Theorem A says also something about the sharpness of Theorem 1.4, indeed itis well known that the power two in (1.11) cannot be replaced by any higher order power(see for example [16]), in particular (1.5) implies that also the power one in (1.12) cannot5e improved. This observation together with (1.13) suggests that Theorem 1.4 is in a sensethe correct generalization of Theorem 1.2.It is worth to mention that another instance (besides Theorem 1.4) where a summabilitycondition on the Gromov-Hausdorff distances is natural and necessary in metric spaces wasalready observed by Colding (see [6, Sec. 4.5]). Roughly said he proves that the summabilityof the Gromov-Hausdorff distance from a cone on dyadic scales on a Riemannian manifoldis necessary and sufficient to have uniqueness of the tangent cone. Moreover he pointsout the discrepancy between this summability assumption in comparison with the squaresummability assumption in Theorem 1.2 by Toro. As for the biLipschitz Reifenberg above,our Theorem A explains this discrepancy.
Acknowledgements
I am grateful to Guido De Philippis and Nicola Gigli for bringing this problem to myattention and for stimulating discussions. I also wish to thank Tatiana Toro and DavidGuy for their feedback and comments during the preparation of this note.
We gather in this section some elementary and well known results that will be needed inthe sequel.In what follows and in all this note we denote by d H the Hausdorff distance betweensets in R d . Moreover given an affine plane Γ in R d and a point x ∈ R d we denote by d ( x, Γ)the distance between x and Γ . The following elementary Lemma is well known in literature (see for example [9, Lemma12.62]).
Lemma 2.1.
For every n ∈ N with n there exists a constant C = C ( n ) > such that thefollowing holds. Suppose that Γ , Γ are two affine n -dimensional planes in R d , with d > n ,such that there exist points { x i } ni =0 ⊂ Γ ∩ B R d (0) satisfying | x i − x − e i | ≤ , for every i = 1 , ..., n, d ( x i , Γ ) ≤ ε, for every i = 0 , ..., n, where e , ..., e d are orthonormal vectors in R d and ε ∈ (0 , / . Then d H (Γ ∩ B (0) , Γ ∩ B (0)) ≤ C ε .
The following result is also standard and can be easily proved applying Gram-Schmidtorthogonalization procedure together with a straightforward computation in coordinates(see for example Lemma 7.11 in [8]).
Lemma 2.2.
For every n ∈ N exists a constant C = C ( n ) > such that the followingholds. Let f : B R n r (0) → R d , d > n , be an ( ε r ) -isometry (see Section 3) with ε ∈ (0 , ,then there exists an isometry I : R n → R d such that I (0) = f (0) and satisfying | I − f | ≤ C √ εr, on B R n r (0) . We now define a notion of distance between affine planes in R d . efinition 2.3. Let Γ , Γ be two affine n -dimensional planes, we put d (Γ , Γ ) := d H (˜Γ ∩ B (0) , ˜Γ ∩ B (0)) , where ˜Γ i is the n dimensional plane parallel to Γ i and passing through the origin.Notice that the function d just defined clearly satisfies d (Γ , Γ ) ≤ d (Γ , Γ ) + d (Γ , Γ ).The following elementary lemma says that if two affine planes are sufficiently close withrespect to the distance d , then they are not orthogonal to each other. Lemma 2.4.
Let Γ , Γ two n -dimensional affine planes in R d such that d (Γ , Γ ) < .Write Γ i as p i + V i where p i ∈ R d and V i is a n -dimensional subspace of R d . Then V ⊥ ⊕ V = R d . In particular for every p ∈ Γ there exists q ∈ Γ such that Π( q ) = p , where Π is theorthogonal projection onto Γ . Proof.
It’s enough to prove that V ⊥ ∩ V = { } . Suppose v ∈ V ⊥ ∩ V , then we can regard V , V as affine planes through the origin and parallel to Γ , Γ . Therefore by hypothesis | v | = d ( v, V ) ≤ d (Γ , Γ ) | v | and thus v = 0 . The following simple technical result will be the main tool for the proof of Theorem 3.1.
Lemma 2.5.
Let Γ , Γ two n -dimensional affine planes in R d . Then for any x ∈ Γ andany y ∈ R d (different from x ) | x − y | ≤ | Π( x ) − Π( y ) | + | x − y | (cid:18) d (Γ , Γ ) + d ( y, Γ ) | x − y | (cid:19) , where Π denotes the orthogonal projection onto Γ .Proof. Let α := d (Γ , Γ ). Up to translating both the plane Γ and the points x, y by thevector Π( x ) − x , we can suppose x ∈ Γ and x = 0. Let now p be the orthogonal projectionof y onto Γ . Since both Γ and Γ contain the origin, we have that d ( p, Γ ) ≤ d H (Γ ∩ B | p | (0) , Γ ∩ B | p | (0)) ≤ | p | α ≤ | y | α. Therefore d ( y, Γ ) ≤ d ( y, p ) + d ( p, Γ ) = d ( y, p ) + d ( p, Γ ) ≤ d ( y, Γ ) + | y | α. Then byPythagoras’ theorem | y | = | Π( y ) − y | + | Π( y ) | = d ( y, Γ ) + | Π( y ) | ≤ ( d ( y, Γ ) + | y | α ) + | Π( y ) − Π( x ) | , since Π( x ) = 0. This concludes the proof.We conclude with two results about the numbers α ( x, r ) and β ( x, r ) (recall their defini-tion in (1.1), (1.6)), which are well known in the literature. The first one shows that thereexists a plane which realizes β ( x, r ) (i.e. that minimizes (1.6)) and at the same time almostrealizes α ( x, r ). The second is a classical tilting estimates, which says that the orientationof such realizing plane do not vary too much from scale to scale and between points closeto each other. 7 roposition 2.6 (Realizing plane) . For every n ∈ N there exists a constant C = C ( n ) ≥ such that the following holds. Let S ⊂ R d with d > n , let x ∈ S and r > be such that α ( x, r ) ≤ / , then there exists an n -dimensional affine plane Γ rx that realizes β ( x, r ) andsuch that r − d H ( S ∩ B r ( x ) , Γ rx ∩ B r ( x )) ≤ Cα ( x, r ) . Proof.
The existence of two planes Γ and Γ ′ that realize respectively β ( x, r ) and α ( x, r ) , follows by compactness. Without loss of generality we can assume that x = 0 and r = 1.Since 0 ∈ Γ ′ there exist orthonormal vectors e , ..., e n ∈ Γ ′ and points x , ..., x n ∈ S ∩ B (0)such that | x i − e i | ≤ α (0 , i = 1 , ..., n. Moreover there exist points y , y , ..., y n ∈ Γ suchthat | y i − x i | , | y | ≤ β (0 , ≤ α (0 , i = 1 , ..., n. In particular | y i − y − e i | ≤ α (0 , i = 1 , ..., n and we can apply Lemma 2.1 to deduce that d H (Γ ∩ B (0) , Γ ′ ∩ B (0)) ≤ Cα (0 , , which concludes the proof. Proposition 2.7 (Tilting estimate) . For any n ∈ N there exist α = α ( n ) , C = C ( n ) > such that the following holds. Let S ⊂ R d with n > d and let r > and x, y ∈ S be suchthat α ( x, r ) , α ( y, r ) ≤ α ( n ) and | x − y | < r . Then it holds d (Γ rx , Γ rx ) ≤ C ( β ( x, r ) + β ( x, r )) , d (Γ rx , Γ ry ) ≤ C ( β ( x, r ) + β ( y, r )) , for any choice of realizing planes Γ rx , Γ ry , Γ rx (as given by Prop. 2.6).Proof. We prove only the second, since the first is analogous.As usual, the scaling and translation invariant nature of the statement allows us toassume that r = 1 and x to be the origin. Then there exist orthonormal vectors e , ..., e n ∈ Γ x and points x = x , x , ..., x n ∈ S ∩ B (0) such that | x i − / e i | ≤ C ( n ) α ( x, , i = 1 , ..., n. Moreover (if α ( x,
1) is small enough) x i ∈ B ( y ), hence there exist points y , ..., y n ∈ Γ y such that | x i − y i | ≤ β ( y, , i = 0 , ..., n. Finally there exist points z , ..., z n ∈ Γ x suchthat | z i − x i | ≤ β ( x, , i = 1 , ..., n. Putting all together we have | y i − y − / e i | ≤ Cα ( x,
1) + 2 β ( y,
1) + β ( x, i = 1 , ..., n and d ( y i , Γ x ) ≤ β (0 ,
1) + β ( y, , i = 0 , ..., n ,hence (if α ( x, , α ( y,
1) are small enough) we can apply Lemma 2.1 to deduce that d H (Γ x ∩ B (0) , Γ y ∩ B (0)) ≤ C ( β ( x,
1) + β ( y, d ( x, Γ y ) ≤ β ( x,
1) + β ( y,
1) andrecalling that x is the origin concludes the proof. a ≤ α and b ≤ β Both Theorem A and Theorem B will be deduced as corollaries of the following more preciseresult.
Theorem 3.1.
For every n ∈ N there exist C = C ( n ) > , ε = ε ( n ) > such that thefollowing holds. Let i ∈ N , S ⊂ R d with d > n and assume that α j ≤ ε for every j ≥ i − (where α j are as in (1.3) ), then a ( x, − i ) ≤ C sup j ∈ N (cid:0) β i − + ... + β i + j (cid:1) j ! ∨ Cα i , ∀ x ∈ S, | x | ≤ − − i , (A) b ( x, − i ) ≤ C sup j ∈ N (cid:0) β i − + ... + β i + j (cid:1) j , ∀ x ∈ S, | x | ≤ − − i , (B) where β j are as in (1.8) . a ( x, r ) ≤ Cα ( x, r ) ” and “ b ( x, r ) ≤ Cβ ( x, r ) ” that are not true in general since, as wesaw in the introduction, (1.2) and (1.7) cannot be improved. Proof of Thoerem A and Thoerem B, given Theorem 3.1.
Let ε ∈ (0 , /
2) and ¯ i ∈ N be asin the hypotheses of Thoerem A and Thoerem B. Since ε > − ¯ i , from Theorem 3.1 and thedefinition of the numbers a i we have( a i ) λ ≤ sup x ∈ S, | x |≤ − − i a ( x, − i ) λ ≤ C sup j ∈ N (cid:0) β i − + ... + β i + j (cid:1) λ λj ! ∨ Cα λi ≤ Cα λi + C X j ≥ ( j + 3) λ − ∨ λj ( β λi − + · · · + β λi + j ) , ∀ i ≥ ¯ i. An analogous estimate holds for b i , ∀ i ≥ ¯ i . Recalling that β i ≤ α i we obtain X i ≥ ¯ i a λi ≤ C X i ≥ ¯ i α λi + C X i ≥ ¯ i X j ≥ ( j + 3) λ − ∨ λj ( α λi − + · · · + α λi + j ) ≤ C X i ≥ ¯ i α λi + C X j ≥ ( j + 3) λ − ∨ λj X i ≥ ¯ i ( α λi − + · · · + α λi + j ) ≤ C X i ≥ ¯ i α λi + C X j ≥ ( j + 3) λ ∨ ( j + 3)2 λj X i ≥ ¯ i − α λi , which proves (1.4). The exact same computations yields also (1.9).Before passing to the proof of Theorem 3.1 we recall the definition of δ -isometry andhow it can be used to estimate the Gromov-Hausdorff distance. Given two metric spaces(X i , d i ), i = 1 , δ > f : X → X is a δ -isometry if | d ( f ( x ) , f ( y )) − d ( x, y ) | < δ for every x, y ∈ X . It holds that d GH ((X , d ) , (X , d )) ≤ { δ > ∃ δ -isometry f : X → X , with f ( X ) δ -dense in X } , see for example [4] for a proof. Proof of Theorem 3.1.
Observe that it is sufficient to consider the case x = 0 and i = 0 forboth (A) and (B), since the conclusion then follows by translating and scaling.We define θ := C sup j ∈ N (cid:0) β − + ... + β j (cid:1) j ,θ ′ := max( θ, Cα ) , where C is a big enough constant depending only on n , to be determined later. Beforeproceeding we make the following observation C ( β − + ... + β j ) > λ > ⇒ θ > λ j . (3.1)Along the proof, for a given x ∈ S and r > rx one of the realizingplanes given by Proposition 2.6 (the choice of the particular plane is not relevant).9 roof of (B) : Let Π be the orthogonal projection onto Γ . It is sufficient to show thatΠ : S ∩ B (0) → Γ ∩ B (0) , is a θ -isometry , (3.2)with respect to the Euclidean distance.Choose x, y ∈ S ∩ B (0) distinct and observe that there exists a unique integer j ≥ j ≤ | x − y | < j − . (3.3)Applying Proposition 2.7 multiple times (assuming α j ≤ α ( n ) for every j ≥ i −
2, with α ( n )as in the statement of Prop. 2.7) we have d (Γ , Γ − j +1 x ) ≤ d (Γ , Γ ) + d (Γ , Γ ) + d (Γ , Γ x ) + d (Γ x , Γ x ) + ... + d (Γ − j +2 x , Γ − j +1 x ) ≤ D ( β (0 ,
0) + β (0 ,
2) + β (0 , ) + β ( x, ) + ... + β ( x, − j +1 )) ≤ D ( β − + ... + β j − ) , (3.4)for some constant D depending only on n . We consider now two cases, when ( D + 4)( β − + ... + β j − ) > C ≥ D + 4, from (3.1) wehave θ ≥ j and therefore || Π( x ) − Π( y ) | − | x − y || ≤ | x − y | ≤ j < θ, that is what we wanted. Hence we can suppose that ( D + 4)( β − + ... + β j − ) ≤ . Sincefrom (3.3) it holds that | x − y | ≥ − j , we have that d ( y, Γ − j +1 x ) ≤ β ( x, − j +1 ) | x − y | . (3.5)We can now apply Lemma 2.5 to the planes Γ , Γ − j +1 x , that coupled with (3.4) and (3.5)gives | Π( x ) − Π( y ) | ≥ | x − y | q − ( D ( β − + ... + β j − ) + 4 β j − ) . Hence | x − y | − | Π( x ) − Π( y ) | ≤ | x − y | (cid:18) − q − (( D + 4)( β − + ... + β j − )) (cid:19) . Thanks to the assumption ( D + 4)( β − + ... + β j − ) ≤
1, we can use the elementary theinequality 1 − √ − x ≤ x , valid for 0 ≤ x ≤
1, to finally obtain || x − y | − | Π( x ) − Π( y ) || ≤ | x − y | (( D + 4)( β − + ... + β j − )) ≤ (( D + 4)( β − + ... + β j − )) j − ≤ θ, where we have used the definition of θ and assuming C ≥ D + 4). This concludes theproof of (3.2) and thus the proof of (B). Proof of (A) : In view of (3.2), we only need to show that Π is also θ ′ -surjective.10 laim: Let C ′ = C ′ ( n ) ≥ p ∈ Γ ∩ B (0) and x ∈ S ∩ B (0) such that C ′ α ≥ | p − Π( x ) | ≥ θ ′ it holds B | p − Π( x ) | ( p ) ∩ Π( S ∩ B (0)) = ∅ . Before proving the claim, we show that it implies that Π is θ ′ -surjective. Indeed supposeit is not, i.e. there exists p ∈ Γ ∩ B (0) such that R := sup { r | B r ( p ) ∩ Π( S ∩ B (0)) = ∅} ≥ θ ′ . Since d H (Γ ∩ B (0) , S ∩ B (0)) ≤ C ′ α (recall that Γ was chosen as a realizing plane asgiven by Prop. 2.6), there exists x ∈ S ∩ B (0) such that | x − p | ≤ C ′ α and in particular | Π( x ) − p | ≤ C ′ α . Therefore R ≤ C ′ α . This implies, from the definition of R , that thereexists a point x ′ ∈ S ∩ B (0) such that θ ′ ≤ R ≤ | Π( x ′ ) − p | ≤ min( R, C ′ α ). However theClaim gives that ∅ 6 = B | Π( x ′ ) − p | ( p ) ∩ Π( S ∩ B (0)) ⊂ B R ( p ) ∩ Π( S ∩ B (0)) , that contradicts the minimality of R . Proof of the Claim : Set R := | p − Π( x ) | . To make the proof more easy to follow we firstexplain the intuition behind it. The key idea is that near x the set S is distributed in ahorizontal manner, near a plane passing through x . We can then move along this planetowards p and thus find a point y in S ∩ B (0) such that | Π( y ) − p | ∼ R . However, since p can be near the boundary of B (0), in this movement we might go outside the ball B (0).To avoid this issue we consider a point q such that | p − q | ∼ R but placed radially towardsthe origin and then find a point y (using the idea described above of moving horizontallynear x ) that projects near q .Start by noticing that (if α is small enough w.r.t. n ) R ≤ C ′ α < / . Therefore thereexists a unique integer j ≥ j +1 ≤ R < j . (3.6)Since by assumption θ ≤ θ ′ ≤ R ≤ / j , from (3.1) we have ( C ( β − + ... + β j )) ≤ / q ∈ Γ ∩ B (0) as q = p − p | p | R . Then | q | = (cid:12)(cid:12)(cid:12)(cid:12) | p | − R (cid:12)(cid:12)(cid:12)(cid:12) ≤ − R , (3.7)indeed | p | < R < . Moreover | p − q | = R/ | q − Π( x ) | ≤ | p − Π( x ) | + | p − q | = 3 / R. Consider the plane Γ − j x , arguing as in (3.4) we can show that d (Γ , Γ − j x ) ≤ C ( β − + ... + β j ) < , provided C is big enough. Then by Proposition 2.4 there exists a point e ∈ Γ − j x such thatΠ( e ) = q . Applying Lemma 2.5 we obtain | e − x | ≤ | q − Π( x ) | + | e − x | / | e − x | ≤ √ | q − Π( x ) | ≤ R ≤ / j − . Therefore there exists y ∈ S ∩ B − j +2 ( x )such that | y − e | ≤ α j − − j +2 < R/ α j − is small enough). Thus | Π( y ) − p | ≤ | Π( y ) − Π( e ) | + | p − q | ≤ | y − e | + R/ < / R, that means Π( y ) ∈ B R ( p ) . It remains to prove that y ∈ B (0) . First we observe that from(3.6) and the assumption R ≤ C ′ α we have | y − x | ≤ j ≤ R ≤ C ′ α . Hence, since x ∈ B (0) , d ( y, Γ ) ≤ | x − y | + d ( x, Γ ) ≤ C ′ α . From previous computations we know that that | Π( y ) − q | = | Π( y ) − Π( e ) | ≤ | y − e | ≤ R/ | Π( y ) | ≤ | q | + R/ ≤ − R/ . From Pythagoras Theorem we obtain | y | = | Π( y ) | + d ( y, Γ ) ≤ (cid:18) − R (cid:19) + (9 C ′ ) α == 1 + R (cid:18) R
16 + (9 C ′ ) α R − (cid:19) . Thus to conclude it is enough to show that R
16 + (9 C ′ ) α R < . Since by assumption R ≥ θ ′ and by definition θ ′ ≥ Cα , we deduce that α R ≤ C . Thereforerecalling that R <
1, the above inequality is satisfied as soon as
C < C ′ ) . This concludesthe proof. α ≤ a , β ≤ b As explained in the introduction and in Section 3, the inequalities “ a ( x, r ) ≤ Cα ( x, r ) ” and“ b ( x, r ) ≤ Cβ ( x, r ) ” are not true in general, but hold only in their weaker formulationscontained in Theorem 3.1. In this final section we will prove that the opposite inequality α ( x, r ) ≤ C a ( x, r ) do hold in general, together with a weaker version of β ( x, r ) ≤ C b ( x, r ). Proposition 4.1.
Let S ⊂ R d and n ∈ N with n < d . Then α ( x, r ) ≤ C a ( x, r ) , for every x ∈ S, r > , where α ( x, r ) is as in (1.1) and C = C ( n ) > . Proof.
Straightforward from Lemma 2.2Given a set of n + 1 points x , ..., x n ∈ R d we denote by Vol n ( x , ..., x n ) the volume ofthe n -dimensional simplex with vertices x , ..., x n . It is well know that for every n ∈ N thereexists a polynomial P n : R ( n +1) n/ → R such that Vol n ( x , ..., x n ) = P n ( {| x i − x j | } ≤ i 40] for a proof.The following results states that β ( x, r ) ≤ C p b ( x, r ), provided B r ( x ) ∩ S contains n points which are ‘sufficiently independent’ in the sense that they span a simplex with largevolume. 12 roposition 4.2. Let S ⊂ R d and n ∈ N with n < d . Then β ( x, r ) ≤ C p b ( x, r ) V n ∧ V n n ! , for every x ∈ S, r > , where V n = sup { x i } ni =0 ⊂ S ∩ B r ( x ) r − n Vol n ( x , ..., x n ) , β ( x, r ) is as in (1.6) and C = C ( n ) > . In particular, if α ( x, r ) < / ( α ( x, r ) is as in (1.1) ), then β ( x, r ) ≤ C p b ( x, r ) , for every x ∈ S, r > , Proof. After a rescaling we can consider only the case r = 1. We can also suppose that V n > , otherwise there is nothing to prove. Fix ε > 0. There exists a map f : S ∩ B (0) → B R n (0) that is a ( b ( x, 1) + ε )-isometry. Moreover there exist { x i } ni =0 ⊂ S ∩ B ( x ) suchthat Vol n ( x , ..., x n ) > V n − ε . Let x n +1 ∈ S ∩ B ( x ) be arbitrary and observer that, since f ( x ) , ..., f ( x n ) , f ( x n +1 ) ∈ R n , we must have Vol n +1 ( f ( x ) , ..., f ( x n ) , f ( x n +1 )) = 0. From(4.1) and the fact that P n +1 is locally Lipschitz, it follows that Vol n +1 ( x , ..., x n , ¯ x ) ≤ C ( n ) sup ≤ i 1) + ε ) . Therefore, denoted by Γ the n -dimensional plane spanned by x , ..., x n , it holds d ( x n +1 , Γ) = Vol n +1 ( x , ..., x n , x n +1 ) Vol n ( x , ..., x n ) ≤ C ( n ) p b ( x, 1) + εV n − ε . Moreover it is clear that there exists a constant C ′ ( n ) > Vol n +1 ( x , ..., x n , x n +1 ) ≤ C ′ V n +1 n n . From the arbitrariness of x n +1 ∈ S ∩ B ( x ) and ε > References [1] J. Azzam, X. Tolsa, Characterization of n-rectifiability in terms of Jones’ squarefunction: Part II, Geom. Funct. Anal. (2015), 1371–1412.[2] C. Bishop, P. Jones, Harmonic measure, L -estimates and the Schwarzian derivative, J. Anal. Math. (1994), 77–113.[3] L. M. Blumenthal, Theory and applications of distance geometry, Second editionChelsea Publishing Co., New York 1970.[4] D. Burago, Y Burago, S. Ivanov, A course in metric geometry, Graduate Studiesin Mathematics , American Mathematical Society, Providence, RI, 2001.[5] J. Cheeger, T. Colding, On the structure of spaces with Ricci curvature boundedbelow. I, J. Differential Geom. , no. 3 (1997), 406-480.[6] T. H. Colding, New monotonicity formulas for Ricci curvature and applications. I, Acta Math. (2012), no. 2, 229–263.[7] G. David, T. De Pauw, T. Toro, A generalization of Reifenberg’s theorem in R , Geom. Funct. Anal. (2008), 1168–1235.138] G. David, T. Toro, Reifenberg flat metric spaces, snowballs, and embeddings, Math.Ann. 315 (1999), no. 4, 641–710.[9] G. David, T. Toro, Reifenberg parameterizations for sets with holes, Mem. Amer.Math. Soc. 215 (2012), no. 1012.[10] N. Edelen, A. Naber, D. Valtorta, Effective Reifenberg theorems in Hilbert andBanach spaces, Math. Ann. (2019), no. 3-4, 1139–1218.[11] P. Jones, Rectifiable sets and the traveling salesman problem, Invent. Math. 102(1990), no. 1, 1–15.[12] A. Lytchak, S. Stadler, Ricci curvature in dimension 2, preprint:arXiv:1812.08225.[13] V. Kapovitch, A. Mondino, On the topology and the boundary of N-dimensional RCD( K, N ) spaces, preprint: arXiv:1907.02614.[14] A. Naber, D. Valtorta, Rectifiable-Reifenberg and the regularity of stationary andminimizing harmonic map, Ann. of Math. (2) 185 (2017), no. 1, 131–227.[15] E. R. Reifenberg, Solution of the Plateau Problem for m -dimensional surfaces ofvarying topological type, Acta Math. (1960), 1–92.[16] T. Toro, Geometric conditions and existence of bi-Lipschitz parameterizations, DukeMath. J.77