Lipschitz Extensions to Finitely Many Points

Abstract We consider Lipschitz maps with values in quasi-metric spaces and extend such maps to finitely many points. We prove that in this context every 1-Lipschitz map admits an extension such that its Lipschitz constant is bounded from above by the number of added points plus one. Moreover, we prove that if the source space is a Hilbert space and the target space is a Banach space, then there exists an extension such that its Lipschitz constant is bounded from above by the square root of the total of added points plus one. We discuss applications to metric transforms.


Introduction
Lipschitz maps are generally considered as an indispensable tool in the study of metric spaces. The need for a Lipschitz extension of a given Lipschitz map often presents itself naturally. Deep extension results have been obtained by Johnson, Lindenstrauss, and Schechtman [JLS86], Ball [Bal92], and Lee and Naor [LN05]. Non linear target spaces have also been studied, see for example [LPS00;MN13]. In [LS05], Lang and Schlichenmaier present a sufficient condition for a pair of metric spaces to have the Lipschitz extension property. The literature surrounding Lipschitz extension problems is vast, for a recent monograph on the subject see [BB11;BB12] and the references therein.
Before we explain our results in detail, we start with a short presentation of what we will call the Lipschitz extension problem.
Let (X, d X ) denote a metric space and let (Y, ρ Y ) be a quasi-metric space, that is, the function ρ Y : Y × Y → R is non-negative, symmetric and vanishes on the diagonal, cf. [Sch38,p. 827]. Unfortunately, the term "quasi-metric space" has several different meanings in the mathematical literature. In the present paper, we stick to the definition given above. Let S ⊂ X be a subset. A Lipschitz map is a map f : S → Y such that the quantity is finite. We use the convention inf ∅ = +∞. We consider the following Lipschitz extension problem: Lipschitz Extension Problem. Let (X, d X ) be a metric space, let (Y, ρ Y ) be a quasi-metric space, and suppose that S ⊂ X is a subset of X. Under what conditions on S, X and Y is there a real number D 1 such that every Lipschitz map f : S → Y has a Lipschitz extension f : X → Y with We use | · | or card(·) to denote the cardinality of a set. The Lipschitz extension modulus e n (X, Y) has been studied intensively in various settings. Nevertheless, many important questions surrounding e n (X, Y) are still open, cf. [NR17] for a recent overview.
In the present article, we are interested in an upper bound for e m (X, Y). We get the following result. Theorem 1.1. Let (X, d X ) be a metric space and let (Y, ρ Y ) be a quasi-metric space. If m 1 is an integer, then e m (X, Y) m + 1.
(1.1) A constructive proof of Theorem 1.1 is given in Section 3.
The estimate (1.1) is optimal. This follows from the following simple example. We set P m+1 := {0, 1, . . . , m, m + 1} ⊂ R and we consider the subset S := Y := {0, m + 1} ⊂ P m+1 and the map f : S → Y given by x → x. Suppose that F : P m+1 → Y is a Lipschitz extension of f to P m+1 . Without effort it is verified that Lip(F) = (m + 1) Lip(f); hence, it follows that (1.1) is sharp. The sharpness of Theorem 1.1 allows us to obtain a lower bound for the parameter α(ω) of the dichotomy theorem for metric transforms [MN11, Theorem 1], see Corollary 2.2.
If the condition that the subset S ⊂ X has to be closed is removed in the definition of e m (X, Y), then Theorem 1.1 is not valid. Indeed, if (X, d X ) is not complete and z ∈ X is a point contained in the completion X of X such that z / ∈ X, then the identity map id X : X → X does not extend to a Lipschitz map id X : X ∪ {z} → X if we equip X ∪ {z} ⊂ X with the subspace metric. This is a well-known obstruction.
As pointed out by Naor and Mendel, there is the following upper bound of e m (X, Y) in terms of e m (X, Y) . Lemma 1.2 (Claim 1 in [MN17]). Let (X, d X ) and (Y, d Y ) be two metric spaces. If m 1 is an integer, then By the use of Lemma 1.2 and [LN05, Theorem 1.10], one can deduce that if (X, d X ) is a metric space and (E, · E ) is a Banach space, then for all integers m 3, where the notation A B means A CB for some universal constant C ∈ (0, +∞). As a result, for sufficiently large integers m 3 the estimate in Theorem 1.1 is not optimal if we restrict the target spaces to the class of Banach spaces.
In Section 4, we present an example that shows that for Banach space targets the estimate (1.1) is sharp if m = 1.
is a stricly-increasing concave function with where m := |X \ S|.
Theorem 1.3 is optimal if m = 1 and F = id, see Proposition 4.1. Via this sharpness result we obtain that certain F-transforms of ℓ p , for p > 2, do not isometrically embed into ℓ 2 , see Corollary 2.1. Let 0 < α 1 and L 0 be real numbers. An (α, L)-Hölder map is a map for all points x, x ′ ∈ X. By considering the function F(x) = x α , with 0 < α 1, we obtain the following direct corollary of Theorem 1.3. Corollary 1.4. Let (H, ·, · H ) be a Hilbert space, let (E, · E ) be a Banach space and let 0 < α 1 and L 0 be real numbers. If X ⊂ H is a finite subset, S ⊂ X, and f : S → E is an (α, L)-Hölder map, then there is an extension f : X → conv(Im(f)) of f such that f is an (α, L )-Hölder map with where m := |X \ S|.
Along the lines of the proof of Claim 1 in [MN17] one can show that if (X, d X ) and (Y, d Y ) are metric spaces, then for all integers m 1 we have  that has a better asymptotic behaviour than estimate (1.4). However, since Lee and Naor use different methods, we believe that our approach has its own interesting aspects.
The paper is structured as follows. In Section 2, we derive some corollaries of our main results. In Section 3 we prove Theorem 1.1 and in Section 4 we show that our extension results are sharp for one point extensions. In [Bal92], Ball introduced the notions of Markov type and Markov cotype of Banach spaces. To establish Theorem 1.3 we estimate quantities that are of similar nature. The necessary estimates are obtained in Section 5 and Section 6. In Section 6, we deal with M-matrices, which appear naturally in the proof of Theorem 1.3. M-matrices have first been considered by Ostrowski, cf. [Ost37], and since then have been investigated in many areas of mathematics, cf. [PB74]. The main result of Section 6, Theorem 6.1, may be of independent interest for the general theory of M-matrices. Finally, a proof of Theorem 1.3 is given in Section 7.

Embeddings and indices of F-transforms
In this section we collect some applications of our main theorems. Let (X, d X ) and (Y, d Y ) be metric spaces and let f : X → Y be an injective map. We set The sharpness of (1.3) if m = 1 allows us to derive a necessary condition for an F-transform of an ℓ p -space to embed into a Hilbert space. x < +∞.
If p ∈ [1, +∞] is an extended real number and where ε ∈ 0, 1 2 , If 2 < p < +∞ is a real number and the F-transform F[ℓ p ] embeds isometrically into a Hilbert space, then We proceed with an application of Theorem 1.1. Let F : [0, +∞) → [0, +∞) be a function with F(0) = 0. Suppose that F is subadditive and strictly increasing. We define for all α 0. Clearly, the function D F : [0, +∞) → [0, +∞) is finite, submutliplicative and non-decreasing. Moreover, for all real numbers x, α 0. The upper index of F is defined by . (2.1) The existence of the limit (2.1) may be deduced via the general theory of subadditive functions, since D F is submultiplicative and non-decreasing, cf.
In [MN11, Theorem 1], Mendel and Naor obtained a dichotomy theorem for the quantity c F (X), if F is concave and non-decreasing. The upper index of F allows us to obtain lower bounds for the rate of growth of c F (P n ), where P n := {0, 1, . . . , n} ⊂ R.
for all n N.
Proof. We may assume that β(F) < 1. Let (Y, ρ Y ) be a quasi-metric space and let (X, d X ) be a metric space. We may employ Theorem 1.1 to conclude that for all m 1. Let ε > 0 be a real number such that α < 1 − β(F) − ε. By the virtue of Theorem 1.2 in [Mal85] there exists a real number C 0 such that for all α C. Consequently, by the use of (2.3) we obtain for all n N := ⌈C⌉ that as desired.
As a consequence of Corollary 2.2, we conclude that if β(F) < 1, then the second possibility of the dichotomy [MN11, Theorem 1] holds. Thus, there is the following natural question: If β(F) = 1, is it true that, then c F (X) = 1 for all finite metric spaces (X, d X )?

Proof of Theorem 1.1
In this section, we derive Theorem 1.1.
We start with a few definitions. Fix ε > 0. Let F ⊂ S be a finite subset such that for each point (3.1) Since S is closed and X \ S is finite, such a set F clearly exists. We set . . , N} → E be a bijective map such that the composition ω • e is a non-decreasing function. We construct the sequence {E ℓ } N ℓ=0 of subsets of E via the following recursive rule: We claim that for each point z ∈ X \ S there exists an integer L z 1 and a unique injective path γ z : {1, . . . , L z } → E N connecting z to a point x z in F. Indeed, the uniqueness part of the claim follows directly, as E N is admissible. Now, we show the existence part. Let z ∈ X\S be a point. Choose an arbitrary point x ∈ F. If the edge {x, z} is contained in E N , then an injective path γ z with the desired property surely exists. Suppose now that {x, z} / ∈ E N . It follows from the recursive construction of E N that in this case there either exists a path in E N from z to x of length greater than or equal to two or there exists a path in E N from z to a point x ′ ∈ F distinct from x. Thus, in any case an injective path γ z with the desired properties exists.
We define the map F ε : X → Y as follows for all z ∈ X \ S.
Let z ∈ X \ S and x ∈ S be points. By the use of the triangle inequality, we compute Let x ′ ∈ F be a point such that the pair (z, x ′ ) satisfies the estimate (3.1). By the recursive construction of E N , it follows that ω(γ z (ℓ)) d(x ′ , z) for all ℓ ∈ {1, . . . , L z }, since the function ω • e is non-decreasing. Hence, by the use of (3.3) we obtain The last inequality follows, since E N is admissible and the paths γ z , γ z ′ are injective; thus, L z + L z ′ m. We have considered all possible cases and we have established that as desired. This completes the proof.

Lower bounds for one point extensions of Banach space valued maps
The collection of examples that we construct in this section is inspired by [Grü60]. We define the sequence {W k } k 0 of matrices via the recursive rule The matrices W k are commonly known as Walsh matrices. For each integer k 1 let W ′ k denote the (2 k − 1) × 2 k matrix that is obtained from W k by deleting the first row of W k . Further, for each integer k 1 and each integer (4.1) ℓ p for all p ∈ [1, +∞] via the canonical embedding. The goal of this section is to prove the following proposition.
Note that Proposition 4.1 implies in particular that e(ℓ 2 , ℓ 1 ) √ 2. The key component in the proof of Proposition 4.1 is the following geometric lemma.
Lemma 4.2. Let k 1 be an integer and suppose that w ∈ R 2 k −1 is a vector such that then it holds that w = 0.
Proof. By the use of a simple induction it is straightforward to show that for all integers k 1. Thus, we may use (4.3) to compute (4.4) By the use of Equations (4.2) and (4.4) we obtain 1. (4.5) As |1 − x| + |−1 − x| 2 for all x ∈ R, the inequality in (4.5) implies that We set r := v (k) 1 1 . Choose a real number ε > 0 such that for each integer ℓ ∈ {1, . . . , 2 k } the intersection ∂B where ·, · R 2 k −1 denotes the standard scalar product on R 2 k −1 . Choose a real number t ∈ (0, 1] such that t w 1 ε. By the choice of t and ε, it follows that −tw, v (4.6) First, suppose that p ∈ [1, +∞). A simple induction implies that two distinct columns of W k are orthogonal to each other. Since the entries of W k consist only of plus and minus one, we obtain that where we use card(·) to denote the cardinality of a set. Hence, if p ∈ [1, +∞), then the identity (4.6) follows. Since the p-norms · p converge pointwise to the maximum norm · ∞ if p → +∞, the identity (4.6) follows also in the case p = +∞, as was left to show. By considering the contraposition of the statement in Lemma 4.2, we may deduce that there is an As a result, we obtain that Hence, it follows that We conclude this section with the proof of Corollary 2.1.
Proof of Corollary 2.1 . Let δ > 0 be a real number. Let k 1 be an integer and let are given as in (4.1) and interpreted as elements of ℓ p via the canonical embedding. It is readily verified that We define the map T : {v (k) Since the map T is a Lipschitz extension of g id , Proposition 4.1 tells us that where 1/q := 1 − 1/p if p = +∞ and 1/q := 1 otherwise. We set γ := A B . Thus, by putting everything together and via a simple scaling argument, we obtain for all x > 0 Thus, since Consequently, as k 1 and δ > 0 are arbitrary, we deduce p The remainder of this section is devoted to calculate the quantity m(x, λ, id, J). Let J ⊂ I be a proper subset. We may suppose that J = 1, . . . , m , where m := card(J). To ease notation, we set λ kℓ := λ(k, ℓ) and we define the matrix Thus, to conclude the proof we calculate the minimum value of the map Φ.
Let U ⊂ H denote the span of the vectors x(k) k∈J c . Clearly, inf Φ| U = inf Φ.
In the following, we compute the minimal value of Φ| U . The subset U ⊂ H is linearly isometric to (R d , · 2 ) for some integer 1 d card(J c ). Consequently, we may suppose (by abuse of notation) for all k ∈ J c that x(k) ∈ R d , say x(k) = (x k1 , . . . , x kd ), and that the function Φ| U : (R d ) m → R is given by the assignment where p i := (p i1 , . . . , p id ) for all integers 1 i m. Using elementary analysis, one can deduce that the minimum value of Φ| U is equal to The following result will play a major role in the proof of Theorem 1.3. The estimate in Theorem 6.1 is sharp. This is the content of the following example.
Example 6.2. Let m 2 be an integer and let M ∈ Mat(m × m; R) be the tridiagonal matrix given by Furthermore, via Jacobi's equality [Jac41], see (6.7), we get for all pairs of integers (i, j) with i = j − 1. By virtue of (6.2) and (6.3) we obtain Consequently, the estimate (6.1) is best possible.
This section is structured as follows. To begin, we gather some information that is needed to prove Theorem 6.1. At the end of the section, we establish Theorem 6.1.
We start with a lemma that calculates the sum in (6.1) if the absolute values from the 2 × 2-minors are removed.
Therefore, the desired equalities follow, since m ij 0 for all distinct integers 1 i, j m.
We proceed with the following corollary. (iii) the matrix M has at least m − 1 zero entries.
Proof. Clearly, item (ii) is a direct consequence of item (i) and item (iii) is a direct consequence of item (ii). To conclude the proof we establish item (i). Lemma 6.3 tells us that Thus, we obtain |m ki | c kk c iℓ = 0 (6.6) for all integers 1 i m. Since each principal submatrix of C is the inverse matrix of an M-matrix, cf. [Joh82, Corollary 3], it follows c kk = 0. Thus, via Equation (6.6) we obtain m ki = 0 or c iℓ = 0 for all i ∈ {1, . . . , m}, as desired.
Theorem 6.1 will be established via a density argument. As it turns out, it will be beneficial to approximate C by matrices with non-vanishing minors. To this end, we need the following genericity condition. Definition 6.5 (generic matrix). Let m 1 be an integer and let A ∈ Mat(m× m; R) be a matrix. Suppose that 1 k m is an integer and let I, J ⊂ {1, . . . , m} be two subsets such that card (I) = card (J) = k.
We use the notation A[I, J] ∈ Mat(k×k; R) to denote the matrix that is obtained from A by keeping the rows of A that belong to I and the columns of A that belong to J. We say that A is generic if The subsequent lemma demonstrates that being generic is a 'generic property' as used in the context of algebraic geometry. We proceed with the following lemma, which is the key component in the proof of Theorem 6.1. Lemma 6.7. Let m 2 and let A ∈ Mat(m × m; R) be a non-negative matrix. If A is a generic matrix, then for all distinct integers 1 k, ℓ m the skew-symmetric matrix A (k,ℓ) ∈ Mat(m × m; R) given by a (k,ℓ) ij := a ik a jℓ − a jk a iℓ , has the property that each two rows of A (k,ℓ) have a distinct number of positive entries.
Proof. We fix two distinct integers 1 k, ℓ m. If m = 2, then each two rows of A (k,ℓ) have a disinct number of positive entries, since A is generic. Now, suppose that m = 3. The matrix A (k,ℓ) is skew-symmetric; hence, as A is generic we obtain that A (k,ℓ) can have 2 3 different sign patterns. If In the following, we show that (6.8) cannot occur. For the sake of a contradiction, we suppose a 12 > 0, we obtain a 1k > a 2k a 1ℓ a 2ℓ . (6.9) Since a (k,ℓ) 31 > 0, we estimate via (6.9) a 3k a 1ℓ > a 1k a 3ℓ > a 2k a 1ℓ a 2ℓ a 3ℓ . (6.10) Thus, (6.10) tells us that a 3k a 2ℓ > a 2k a 3ℓ ; which contradicts a

31
> 0 cannot occur. The other invalid sign pattern can be treated analogously . Therefore, (6.8) cannot occur, as claimed. By putting everything together, we conclude that the statement is valid if m = 3.
We proceed by induction. Let m 4 be an integer and suppose that the statement is valid for all 2 m ′ < m.
Before we proceed with the proof we introduce some notation. For every matrix B ∈ Mat(m × m; R) we denote by B ij ∈ Mat((m − 1) × (m − 1); R) the matrix that is obtained from B by deleting the i-th row and the j-th column of B. Moreover, for all integers 1 i, j m with i = j we set We use b ij to indicate that the entry b ij is omitted.
Since the non-negative (m − 1) × (m − 1)-matrix A ij is generic for all 1 i, j m, we obtain via the induction hypothesis that each row of A (k,ℓ) ii has a different number of positive entries for all 1 i m.
For simplicity of notation, we abbreviate B := A (k,ℓ) for the rest of this proof. We have to show that each two rows of B have a distinct number of positive entries.
Let p ∈ {1, . . . , m} \ {m} denote the unique integer such that n + p,m (B) = (m − 1) − 1, that is, the p-th row of B mm has the most positive entries. Suppose that b pm > 0. This implies n + p (B) = m − 1. Consequently, the p-th column of B has no positive entries; hence, as each two rows of B pp have a distinct number of positive entries and the number of positive entries of each row of B pp is strictly smaller than m − 1, we obtain that all rows of B have a distinct number of positive entries. Hence, the statement follows if b pm > 0. Now, we suppose that b pm < 0. This implies n + p (B) = m − 2. There is precisely one integer q ∈ {1, . . . , m} \ {p} such that n + q,p (B) = (m − 1) − 1.
Suppose that q = m. Since b mp > 0, we obtain that n + m (B) = m − 1.
Thus, we obtain as before via the induction hypothesis that all rows of B have a distinct number of positive entries. Therefore, the statement follows if q = m.  Therefore, by the use of (6.14) we obtain Proof of Theorem 1.3. Without loss of generality we may assume (by scaling) that Lip(f) = 1. We set I := X, T := X \ S and let the map x : I → H be given by the identity.

Lemma 6.3 tells us that
Let G : [0, +∞) → [0, +∞) denote the function such that x = F( G(x)) for all real numbers x ∈ [0, +∞). Observe that the function G is convex, strictlyincreasing and G(0) = 0. We say that ξ : I × I → R lies above f if there is a map f : X → conv(Im(f)) such that f(s) = f(s) for all s ∈ S and G f x(i) − f x(j) H ξ(i, j) for all i, j ∈ I.
We use conv to denote the closed convex hull. Let E f ⊂ R I×I be the set of all ξ ∈ R I×I that lie above f. Moreover, let v : I × I → R be the map given by v(i, j) := x(i) − x(j) 2 H . (7.1) Suppose that L ∈ [1, +∞) is a real number. If Lv ∈ E f , then the map f admits a Lipschitz extension f : X → E such that .

Indeed, if
Lv ∈ E f , then (by definition) there exists a function f : X → conv(Im(f)) such that G f x(i) − f x(j) H Lv(i, j) for all i, j ∈ I; consequently, by applying the function F ( √ ·) on both sides, we obtain To this end, we suppose that Lv / ∈ E f and we show that L < (m + 1). Since the function G is strictly-increasing and convex, the set E f is closed and convex; thus, by the hyperplane separation theorem we obtain a real number ε > 0 and a non-zero vector λ ∈ R I×I such that Lv, λ R I×I + ε < ξ, λ R I×I for all ξ ∈ E f . (7.2) We claim that each entry of λ is non-negative. Indeed, if ξ ∈ E f , then the point (ξ 1 , . . . , ξ k−1 , cξ k , ξ k+1 , . . . , ξ N ), where N := card(I × I), is contained in E f for all integers 1 k N and real numbers c ∈ [1, +∞). Hence, a simple scaling argument implies that the k-th entry of λ is non-negative for each integer 1 k N, as claimed.
In the following, we estimate Lv, λ R I×I from below. We may assume that λ is symmetric. By adjusting ε > 0 if necessary, we may assume that k∈S λ ik = 0 for all i ∈ T . Let the matrix M := M(λ, T ) be given as in (5.1). Since each entry of the vector λ is non-negative and k∈S λ ik = 0 for all i ∈ T , the matrix M(λ, T ) is non-singular. We set C := M −1 . Proposition 5.1 tells us that

Lm
Lv, λ R I×I . (7.4) Next, we estimate Lv, λ R I×I from above. We set Using (7.5) we obtain w i ∈ conv(Im(f)) for all i ∈ T . Equation (7.2) tells us that Lv, λ R I×I < A + B + C;