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LEVEL 1 OF 1 · Metric Markov cotype two for $\ell_1$
Metric Markov Cotype Two of ℓ1
expertly designed by an internal OpenAI model · released 2026-10-05
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IntroductionMetric Markov cotype asks whether the values of a map on a finite reversible Markov chain can be modified so that their one-step variation becomes small, while the cost of the modification is controlled by the original variation over a longer time interval. The modified values are not prescribed linear averages. This freedom is essential for the result proved here. We work first with the real Banach space \[\ell_1=\left\{x=(x(k))_{k\geq1}:\sum_{k\geq1}|x(k)|<\infty\right\}, \qquad \|x\|_1=\sum_{k\geq1}|x(k)|.\] Write \(\mathbb N=\{1,2,\ldots\}\) and \([n]=\{1,\ldots,n\}\). A stochastic matrix \(A=(a_{ij})_{i,j\in[n]}\) has nonnegative entries and each row sum equal to one. It is reversible with respect to a probability vector \(\pi=(\pi_i)_{i\in[n]}\) if \(\pi_i a_{ij}=\pi_j a_{ji}\) for all \(i,j\). Reversibility implies stationarity, \(\pi A=\pi\). Zero coordinates of \(\pi\) are allowed. Definition 1 (Mendel–Naor). A metric space \((X,d)\) has metric Markov cotype two with constant \(C>0\) if, for every \(n,t\in\mathbb N\), every probability vector \(\pi\), every stochastic matrix \(A\) reversible with respect to \(\pi\), and every \(x_1,\ldots,x_n\in X\), there exist \(y_1,\ldots,y_n\in X\) such that \[ \sum_i\pi_i d(x_i,y_i)^2 +t\sum_{i,j}\pi_i a_{ij}d(y_i,y_j)^2 \leq C^2\sum_{i,j}\pi_i \left(\frac1t\sum_{s=1}^t A^s\right)_{ij}d(x_i,x_j)^2. \tag{1}\] The infimum of all such constants is denoted \(N_2(X)\). This is (Mendel and Naor 2013, Definition 1.2). Our main result is the following. Theorem 2. The real space \(\ell_1\) satisfies \[N_2(\ell_1)\leq12\sqrt{21}.\] Thus the square of the constant in (1) may be taken to be \(3024\), independently of the number of points, the number of coordinates, and the time parameter. No optimization of this numerical constant is intended. History and significanceBall introduced Markov type and cotype in his study of Lipschitz extension (Ball 1992). His Markov cotype condition for Banach spaces uses specified linear averages, and \(\ell_1\) fails that stronger condition. Ball also proposed a metric definition. Mendel and Naor developed the Cesàro formulation above, proved its equivalence to Ball’s metric definition, and asked explicitly whether \(N_2(\ell_1)\) is finite (Mendel and Naor 2013, Question 1.15 and Section 7). Theorem 2 answers that question affirmatively. For \(1<p\leq2\), the uniformly convex theory gives \(N_2(\ell_p)\leq C/\sqrt{p-1}\) with a universal constant \(C\) (Ball 1992); see also (Mendel and Naor 2013, sec. 1.1). These bounds deteriorate as \(p\) decreases to one and do not settle the endpoint treated here. There is a second distinction worth making. Using a construction of Kalton, Mendel and Naor found a closed linear subspace of \(\ell_1\) that fails metric Markov cotype at every finite exponent (Mendel and Naor 2013, Theorem 1.14). This does not rule out Theorem 2, because the points \(y_i\) in Definition 1 need not belong to a given subspace containing the original data. Our construction uses this freedom. One important consequence concerns extension of Lipschitz maps. For a map \(f\) between metric spaces, \(\mathop{\mathrm{Lip}}(f)\) denotes the least \(L\geq0\) such that \(d(f(x),f(y))\leq Ld(x,y)\) for all \(x,y\) in its domain. Ball’s extension problem asks whether a Lipschitz map from an arbitrary subset of Hilbert space into \(\ell_1\) admits an extension with a universal multiplicative loss in its Lipschitz constant. Mendel and Naor explain that an affirmative answer to their metric Markov cotype question gives such an extension (Mendel and Naor 2013, Corollary 1.13 and the discussion after Question 1.15). The extension problem is also discussed in (Naor and Young 2022, sec. 1). We record the consequence precisely in Section 6; the proof of Theorem 2 itself is elementary and independent of extension theorems. The nonlinear constructionFinite subsets of \(\ell_1\) admit exact representations by weighted cuts; see (Deza and Laurent 1997). We retain not only their distances but also a contractive map back to the given ambient \(\ell_1\). Nonlinear cut smoothing with the cubic \(\phi(r)=3r^2-2r^3\) also appears in the work of Cheng, Wang, and Xiang on sharp metric cotype (Cheng et al. 2026); metric cotype and metric Markov cotype are different invariants. The additional ingredient here is a martingale estimate for the cubic. For a finite family \(\mathcal B\) of cuts, the representation gives binary vectors \(z_i\in\{0,1\}^{\mathcal B}\) in a weighted Hilbert space \(H\), with \[\|z_i-z_j\|_H^2=\|x_i-x_j\|_1.\] It also gives a map \(T\) with \(T(z_i)=x_i\) that is contractive for the weighted \(\ell_1\) norm on the cut coordinates. Let \(Q=[0,1]^{\mathcal B}\), and let \(\Phi:Q\to Q\) apply \(\phi\) coordinatewise. Stop an \(A\)-walk \((X_s)\) after an independent nonnegative geometric number \(S\) of steps with mean \(t\). Writing \(\mathbb E_i\) for expectation when the walk starts at \(i\), we choose \[h_i=\mathbb E_i z_{X_S},\qquad y_i=T(\Phi(h_i)).\] For the estimate, start the walk from its stationary law \(\pi\) and record stopping as a separate transition after the \(S\) walk steps. Attach \(h_j\) to the current index \(j\) while alive and \(z_j\) when the walk stops at \(j\), keeping that value thereafter. These values form a martingale. Because \(\phi'\) vanishes at the binary values, a fourth-power Hilbert potential controls the total squared weighted-\(\ell_1\) variation after applying \(\Phi\). Centering at \(z_{X_0}\) makes the terminal potential equal to \(\|x_{X_S}-x_{X_0}\|_1^2\). The stopping jump accounts for the approximation cost, and the expected \(t\) preceding jumps account for the one-step variation. Contractivity of \(T\) transfers these bounds to \(\ell_1\). We prove the cut representation in Section 2 and the martingale estimate in Section 3. The comparison in Section 4, valid in any metric space, converts the terminal geometric endpoint cost to the Cesàro average in (1). These ingredients are assembled in Section 5. The resulting points have a simple interpretation: each \(y_i\) is the expected coordinatewise median of three independent geometric-walk endpoints started at \(i\). This interpretation is established after the main proof. Finite cuts and reconstructionWe first replace finitely many vectors in \(\ell_1\) by finitely many binary coordinates. The reconstruction map will allow us to choose new points in the full ambient space, rather than in the span of the data. This is the cut representation (Deza and Laurent 1997, chap. 4), together with the ambient reconstruction given below. Lemma 3. Given \(x_1,\ldots,x_n\in\ell_1\), either all the points coincide, or there exist a finite set \(\mathcal B\), positive weights \((w_B)_{B\in\mathcal B}\), binary vectors \(z_i\in\{0,1\}^{\mathcal B}\), and an affine map \(T:\mathbb R^{\mathcal B}\to\ell_1\) with the following properties. For \[\|u\|_{1,w}=\sum_{B\in\mathcal B}w_B|u_B|, \qquad \|u\|_H^2=\sum_{B\in\mathcal B}w_Bu_B^2,\] one has \[\begin{align*} T(z_i)&=x_i,\tag{2}\\ \|x_i-x_j\|_1 &=\|z_i-z_j\|_{1,w}=\|z_i-z_j\|_H^2, \tag{3}\\ \|T(u)-T(u')\|_1&\leq\|u-u'\|_{1,w} \qquad(u,u'\in\mathbb R^{\mathcal B}). \tag{4}\end{align*}\] The space \(H\) is \(\mathbb R^{\mathcal B}\) with the indicated Hilbert norm. Proof. For each original coordinate \(k\), put \(v(k)=\min_i x_i(k)\). For every nonempty proper subset \(B\subset[n]\), define the nonnegative vector \[ b_B(k)=\left(\min_{i\in B}x_i(k) -\max_{i\notin B}x_i(k)\right)_+. \tag{5}\] Here \(r_+=\max\{r,0\}\). The estimates \[|v(k)|\leq\sum_i|x_i(k)|, \qquad b_B(k)\leq2\sum_i|x_i(k)|\] show that \(v,b_B\in\ell_1\). Set \(w_B=\|b_B\|_1\) and retain only subsets with \(w_B>0\); these form \(\mathcal B\). There are at most \(2^n-2\) of them. Define \[ z_i(B)=\mathbf 1_{\{i\in B\}},\qquad T(u)=v+\sum_{B\in\mathcal B}u_B b_B. \tag{6}\] The sum is finite, so \(T\) takes values in \(\ell_1\). To verify the identities, fix \(k\) and list the distinct values among \(x_1(k),\ldots,x_n(k)\) as \(\alpha_1<\cdots<\alpha_m\). For \(1\leq r<m\), the upper cut \[B_r=\{i:x_i(k)\geq\alpha_{r+1}\}\] has \(b_{B_r}(k)=\alpha_{r+1}-\alpha_r\); all other cuts have \(b_B(k)=0\). Thus \(T(z_i)(k)\) is \(\alpha_1\) plus exactly the successive gaps below \(x_i(k)\), proving (2). Moreover, the cuts with positive \(b_B(k)\) are nested. If \(x_i(k)\geq x_j(k)\), every nonzero term in \(\sum_B(z_i(B)-z_j(B))b_B(k)\) is nonnegative; the reverse inequality gives the reverse sign. Hence \[|x_i(k)-x_j(k)| =\sum_B |z_i(B)-z_j(B)|b_B(k).\] Sum over \(k\). Since binary differences have equal absolute value and square, this gives (3). The triangle inequality applied to (6) gives (4). If \(\mathcal B\) is empty, the same coordinate description shows that all the data coincide. ◻ A flat map and a quartic potentialBy (3), squared \(\ell_1\) distances between the original data are fourth powers of Hilbert distances between their binary encodings. We therefore seek a fourth-power potential that controls squared increments after the cubic map. Fix a finite nonempty set \(\mathcal B\) and positive weights \(w_B\), and use the two norms from Lemma 3. Write \(Q=[0,1]^{\mathcal B}\) and let \(\langle\cdot,\cdot\rangle_H\) be the weighted Hilbert inner product. Define \[\phi(r)=3r^2-2r^3\quad(0\leq r\leq1), \qquad \Phi(u)_B=\phi(u_B)\quad(u\in Q).\] The map \(\Phi\) takes \(Q\) to itself and fixes its binary vertices. The key estimate controls its squared weighted-\(\ell_1\) increment by a convexity remainder in \(H\). Lemma 4. For \(u,u'\in Q\) and \(e\in\{0,1\}^{\mathcal B}\), put \(a=u-e\), \(\delta=u'-u\), and \(F_e(u)=\|u-e\|_H^4\). Then \[ \|\Phi(u')-\Phi(u)\|_{1,w}^2 \leq108\bigl(F_e(u')-F_e(u) -4\|a\|_H^2\langle a,\delta\rangle_H\bigr). \tag{7}\] Proof. For either bit \(d\in\{0,1\}\) and \(r\in[0,1]\), \[|\phi'(r)|=6r(1-r)\leq6|r-d|.\] Integrating on the segment from \(u_B\) to \(u'_B\) gives \[\begin{align*} |\phi(u'_B)-\phi(u_B)| &\leq6|\delta_B|\int_0^1|a_B+s\delta_B|\,ds\\ &\leq6|a_B||\delta_B|+3|\delta_B|^2. \end{align*}\] Summing with weights and applying Cauchy–Schwarz yields \[ \|\Phi(u')-\Phi(u)\|_{1,w} \leq6\|a\|_H\|\delta\|_H+3\|\delta\|_H^2. \tag{8}\] Let \(R\) denote the expression in parentheses in (7). Expanding the fourth power gives the exact identity \[ R=2\|a\|_H^2\|\delta\|_H^2 +\bigl(2\langle a,\delta\rangle_H+\|\delta\|_H^2\bigr)^2. \tag{9}\] In particular, \(\|a\|_H^2\|\delta\|_H^2\leq R/2\). Also \[\begin{align*} \|\delta\|_H^4 &\leq2\bigl(2\langle a,\delta\rangle_H+\|\delta\|_H^2\bigr)^2 +8\langle a,\delta\rangle_H^2\\ &\leq2\bigl(2\langle a,\delta\rangle_H+\|\delta\|_H^2\bigr)^2 +8\|a\|_H^2\|\delta\|_H^2 \leq4R. \end{align*}\] Squaring (8) and using \((r+s)^2\leq2r^2+2s^2\) therefore gives \[\|\Phi(u')-\Phi(u)\|_{1,w}^2 \leq72\|a\|_H^2\|\delta\|_H^2+18\|\delta\|_H^4 \leq36R+72R=108R.\] ◻ The linear term in (7) disappears on taking a conditional expectation along a Hilbert-valued martingale. It is important that the center may depend on the initial state. Corollary 5. Let \((M_m)_{m\geq0}\) be an \(H\)-valued martingale with respect to a filtration \((\mathcal F_m)\), taking values in \(Q\) and converging almost surely to \(M_\infty\). If \(e\) is an \(\mathcal F_0\)-measurable binary vector, then \[\begin{align*} \mathbb E\sum_{m=0}^\infty \|\Phi(M_{m+1})-\Phi(M_m)\|_{1,w}^2 &\leq108\bigl(\mathbb E\|M_\infty-e\|_H^4 -\mathbb E\|M_0-e\|_H^4\bigr)\\ &\leq108\mathbb E\|M_\infty-e\|_H^4. \tag{10}\end{align*}\] Proof. Apply Lemma 4 to each pair \(M_m,M_{m+1}\). The vector \(4\|M_m-e\|_H^2(M_m-e)\) is \(\mathcal F_m\)-measurable and bounded. Therefore \[\mathbb E\bigl[\|M_m-e\|_H^2 \langle M_m-e,M_{m+1}-M_m\rangle_H\mid\mathcal F_m\bigr]=0.\] Taking expectations and summing over \(0\leq m<L\) telescopes the potentials. All potentials are bounded by \((\sum_Bw_B)^2\), so bounded convergence applies to the terminal potential as \(L\to\infty\). Monotone convergence applies to the nonnegative jump sum. This proves the first inequality; the second follows by discarding the initial nonnegative potential. ◻ Geometric and Cesàro endpoint costsThe stopped martingale will produce a geometric average of endpoint distances. The next estimate converts it to the finite average in Definition 1. This comparison is part of the resolvent approach in (Mendel and Naor 2013, sec. 7, especially Lemma 7.1). We give an elementary version with an explicit constant; only stationarity, not reversibility, is needed. Lemma 6. Let \((X_s)_{s\geq0}\) be a stationary Markov chain on \([n]\), and let \(x_1,\ldots,x_n\) lie in a metric space \((X,d)\). For \(t\in\mathbb N\), set \(p=1/(t+1)\) and \(q=1-p\). If \(S\) is independent of the walk and \(\mathbb P(S=s)=pq^s\) for \(s\geq0\), then \[ \mathbb Ed(x_{X_S},x_{X_0})^2 \leq\frac{28}{t}\sum_{s=1}^t \mathbb Ed(x_{X_s},x_{X_0})^2. \tag{11}\] Proof. Write \[D(s)=\mathbb Ed(x_{X_s},x_{X_0})^2, \qquad W=\frac1t\sum_{s=1}^t D(s).\] The triangle inequality, Minkowski’s inequality in \(L_2\), and stationarity imply \[ \sqrt{D(r+s)}\leq\sqrt{D(r)}+\sqrt{D(s)} \qquad(r,s\geq0). \tag{12}\] Consequently \(D(t)\leq2D(r)+2D(t-r)\) for \(1\leq r\leq t\). Since \(D(0)=0\), averaging gives \[ D(t)\leq\frac2t\sum_{r=1}^tD(r) +\frac2t\sum_{r=0}^{t-1}D(r)\leq4W. \tag{13}\] If \(s=jt+r\) with \(j\geq0\) and \(0\leq r<t\), repeated use of (12) gives \[ D(s)\leq\bigl(j\sqrt{D(t)}+\sqrt{D(r)}\bigr)^2 \leq2j^2D(t)+2D(r). \tag{14}\] For \(J=\lfloor S/t\rfloor\) and \(R=S-tJ\), the geometric second moment yields \[\mathbb EJ^2\leq\frac{\mathbb ES^2}{t^2} =\frac{2t^2+t}{t^2}\leq3.\] For each \(0\leq r<t\), \[\mathbb P(R=r)=\frac{pq^r}{1-q^t}\leq\frac2t.\] Indeed \((1+1/t)^t\geq2\) implies \(q^t\leq1/2\), so the left side is at most \(2p=2/(t+1)\leq2/t\). Thus \[\mathbb ED(R)\leq\frac2t\sum_{r=0}^{t-1}D(r)\leq2W.\] Take expectations in (14) and use independence of \(S\) and the walk to conclude \[\mathbb Ed(x_{X_S},x_{X_0})^2=\mathbb ED(S) \leq2\cdot3\cdot4W+2\cdot2W=28W.\] Every step remains valid when \(t=1\); then \(R=0\). ◻ The stopped walk and the main theoremWe now construct the required points. Fix the data in Definition 1 with \(X=\ell_1\). If all \(x_i\) coincide, choose \(y_i=x_i\). Otherwise use Lemma 3 to obtain \(z_i,H,T\), and use the map \(\Phi\) of Section 3. Put \(p=1/(t+1)\), \(q=1-p\), and define \[ h_i=p\sum_{s=0}^\infty q^s(A^s z)_i, \qquad (A^s z)_i=\sum_j(A^s)_{ij}z_j. \tag{15}\] This convex combination converges in the finite-dimensional space \(H\) and belongs to \(Q\). Equivalently, \(h_i=\mathbb E_i z_{X_S}\), where \(\mathbb E_i\) refers to an \(A\)-walk started at \(i\) and \(S\) is independent with \(\mathbb P(S=s)=pq^s\). The first-step identity is \[ h_i=pz_i+q\sum_j a_{ij}h_j. \tag{16}\] We take \[ y_i=T(\Phi(h_i)). \tag{17}\] In particular, each \(y_i\) belongs to \(\ell_1\). A martingale that realizes the two costsStart a chain in an alive state with index distributed according to \(\pi\). From an alive state of index \(i\), it moves to the dead state of index \(i\) with probability \(p\), and to the alive state of index \(j\) with probability \(qa_{ij}\). Dead states are absorbing. Its filtration \(\mathcal F_m\) records the initial state and the first \(m\) transitions only. One realization uses an independent stationary \(A\)-walk \((X_s)_{s\geq0}\) and successive independent survival trials, each successful with probability \(q\). Let \(S\) be the number of successful trials before the first failure. Then \(\mathbb P(S=s)=pq^s\) and \(S\) is independent of the walk. There are \(S\) live index transitions followed by one death transition; the index at death is \(X_S\). Attach the value \(h_i\) to the alive state of index \(i\) and the value \(z_i\) to its dead state, and let \(M_m\) be the attached value at chain time \(m\). Equation (16) shows that \((M_m)\) is an \(H\)-valued martingale for the indicated filtration. It lies in \(Q\) and is eventually constant almost surely, with \[M_0=h_{X_0},\qquad M_\infty=z_{X_S}.\] The vector \(e=z_{X_0}\) is \(\mathcal F_0\)-measurable and binary. Corollary 5 and (3) give \[ \mathbb E\sum_{m=0}^\infty \|\Phi(M_{m+1})-\Phi(M_m)\|_{1,w}^2 \leq108\mathbb E\|x_{X_S}-x_{X_0}\|_1^2. \tag{18}\] To compute the left side exactly, set \[B=\sum_i\pi_i\|z_i-\Phi(h_i)\|_{1,w}^2, \qquad E_A=\sum_{i,j}\pi_i a_{ij} \|\Phi(h_i)-\Phi(h_j)\|_{1,w}^2.\] At time \(m\), the event that the chain is alive has probability \(q^m\) and is independent of \(X_m\). Stationarity gives \(\mathbb P(X_m=i)=\pi_i\) and \(\mathbb P(X_m=i,X_{m+1}=j)=\pi_i a_{ij}\). Since \(\Phi(z_i)=z_i\), the expected squared jump at time \(m\) is \(pq^m B+q^{m+1}E_A\). Therefore \[ \mathbb E\sum_{m=0}^\infty \|\Phi(M_{m+1})-\Phi(M_m)\|_{1,w}^2 =B+\frac qp E_A=B+tE_A. \tag{19}\] This identity explains why the killing probability was chosen to be \(1/(t+1)\). Completion of the proofProof of Theorem 2. Use the points (17). Contractivity of \(T\) and the identity \(T(z_i)=x_i\) imply \[\begin{align*} &\sum_i\pi_i\|x_i-y_i\|_1^2 +t\sum_{i,j}\pi_i a_{ij}\|y_i-y_j\|_1^2\\ &\hspace{25mm}\leq B+tE_A \leq108\mathbb E\|x_{X_S}-x_{X_0}\|_1^2, \end{align*}\] where the last inequality follows from (18) and (19). Lemma 6 bounds the last expression by \[108\cdot28\cdot\frac1t\sum_{s=1}^t \mathbb E\|x_{X_s}-x_{X_0}\|_1^2.\] The stationary walk satisfies \[\frac1t\sum_{s=1}^t\mathbb E\|x_{X_s}-x_{X_0}\|_1^2 =\sum_{i,j}\pi_i\left(\frac1t\sum_{s=1}^t A^s\right)_{ij} \|x_i-x_j\|_1^2.\] This is (1) with \(C^2=3024\). ◻ Remark 7. The proof uses reversibility only through \(\pi A=\pi\). Consequently, the same inequality and the same construction hold for every stationary stochastic matrix, without detailed balance. No positivity assumption on the individual \(\pi_i\) is needed in the argument. An expected-median interpretationThe construction has an intrinsic description in the original coordinates. Let \[G=p\sum_{s=0}^\infty q^s A^s\] be the stochastic matrix of geometric endpoints. For fixed \(i\), take independent indices \(J_1,J_2,J_3\) with common law \(\mathbb P(J_a=j)=G_{ij}\). For three real vectors, write \(\mathop{\mathrm{med}}(x,x',x'')\) for their coordinatewise median. This vector belongs to \(\ell_1\), since its coordinatewise absolute value is at most \(|x|+|x'|+|x''|\). Then \[ y_i=\mathbb E\,\mathop{\mathrm{med}}(x_{J_1},x_{J_2},x_{J_3}). \tag{20}\] In each original coordinate, the median is at least a given threshold exactly when at least two of the three values are. The threshold decomposition in the proof of Lemma 3 therefore gives the pointwise identity \[\mathop{\mathrm{med}}(x_{J_1},x_{J_2},x_{J_3}) =v+\sum_{B\in\mathcal B} b_B \mathbf 1_{\{\#\{a\in[3]:J_a\in B\}\geq2\}}.\] Each indicator has expectation \(3h_i(B)^2(1-h_i(B))+h_i(B)^3=\phi(h_i(B))\). Taking expectations in the finite sum proves (20). Complex scalars and Lipschitz extensionCorollary 8. For complex scalars, \(N_2(\ell_1(\mathbb C))\leq12\sqrt{42}\). Proof. The real-linear bijection \[J:\ell_1(\mathbb C)\longrightarrow\ell_1(\mathbb R),\qquad Jx=(\Re x(1),\Im x(1),\Re x(2),\Im x(2),\ldots)\] satisfies \(\|x\|_1\leq\|Jx\|_1\leq\sqrt2\|x\|_1\). Apply Theorem 2 to \(Jx_i\) and pull the resulting points back by \(J^{-1}\). The left side of (1) can only decrease, while its right side increases by at most a factor of \(2\). ◻ For the extension consequence, recall that a metric space \((Z,d_Z)\) has Markov type two with constant \(M>0\) if \[\sum_{i,j}\pi_i(A^t)_{ij}d_Z(u_i,u_j)^2 \leq M^2t\sum_{i,j}\pi_i a_{ij}d_Z(u_i,u_j)^2\] for every \(n,t\in\mathbb N\), every stochastic matrix \(A\) reversible with respect to a probability vector \(\pi\), and every \(u_1,\ldots,u_n\in Z\). Hilbert spaces satisfy this inequality with \(M=1\) (Ball 1992); see also (Mendel and Naor 2013, Definition 1.1 and the following discussion). Corollary 9. There is a universal constant \(K<\infty\) such that, for every real Hilbert space \(\mathcal H\), every subset \(S\subseteq\mathcal H\), and every Lipschitz map \(f:S\to\ell_1\), there is an extension \(F:\mathcal H\to\ell_1\) with \(\mathop{\mathrm{Lip}}(F)\leq K\mathop{\mathrm{Lip}}(f)\). Proof. We use the Lipschitz extension theorem in (Mendel and Naor 2013, Corollary 1.13), extending Ball’s theorem (Ball 1992). In its Banach-space form, a source with Markov type two and a dual Banach target with metric Markov cotype two admit extensions with a constant bounded by a universal multiple of the two corresponding constants. A Hilbert space has Markov type two with constant one, and \(\ell_1\) is the dual of \(c_0\). Theorem 2 verifies the remaining hypothesis. ◻
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Cheng, Qingjin, Yue Wang, and Bo Xiang. 2026. Sharp Metric Cotype Inequalities for \(L_1\) via Nonlinear Cut Smoothing. https://doi.org/10.48550/arXiv.2609.08749.
Deza, Michel Marie, and Monique Laurent. 1997. Geometry of Cuts and Metrics. Vol. 15. Algorithms and Combinatorics. Springer-Verlag. https://doi.org/10.1007/978-3-642-04295-9.
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Naor, Assaf, and Robert Young. 2022. “Foliated Corona Decompositions.” Acta Mathematica 229 (1): 55–200. https://doi.org/10.4310/ACTA.2022.v229.n1.a2.
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