Pythagorean powers of hypercubes (Q332213)

For a metric space \((X,d_X)\), the Cartesian power \(X^n\) is considered as a metric space with the distance between the \(n\)-tuples \((x_1,\dots,x_n)\) and \((y_1,\dots,y_n)\in X^n\) given as NEWLINE\[NEWLINE\sqrt{d_X(x_1,y_1)^2+\dots+d_X(x_n,y_n)^2}.NEWLINE\]NEWLINE The obtained metric space is denoted by \(\ell_2^n(X)\). Similarly, one can define \(\ell_p^n(X)\). If \(X\) is a Banach space, then the space \(\ell_p^n(X)\) has a natural Banach-space structure.NEWLINENEWLINEThe following inequality is due to \textit{S.~Kwapien} and \textit{C.~Schütt} [Stud. Math. 95, No. 2, 141--154 (1989; Zbl 0706.46014)]: There exists an absolute constant \(C\) such that, for every \(n\in\mathbb{N}\) and every \(\{z_{jk}\}_{j,k\in \{1,\dots, n\}}\subset L_1\),NEWLINENEWLINENEWLINE\[NEWLINE\frac{1}{n}\sum_{j=1}^n\sum_{\varepsilon\in \{-1,1\}^n}\Big\|\sum_{k=1}^n\varepsilon_kz_{jk}\Big\|_1 \leq \frac{C}{n!} \sum_{\pi\in S_n} \sum_{\varepsilon\in \{-1,1\}^n} \Big\|\sum_{j=1}^n\varepsilon_jz_{j\pi(j)}\Big\|_1, NEWLINE\]NEWLINE where \(S_n\) denotes the group of all permutations of \(\{1,\dots, n\}\). This inequality implies that \(||T||\cdot||T^{-1}||\geq\sqrt{n}/C\) for every linear injective embedding \(T:\ell_2^n(\ell_1^n)\to L_1\).NEWLINENEWLINEIn the spirit of the Ribe program (see [\textit{A. Naor}, Jpn. J. Math. (3) 7, No. 2, 167--233 (2012; Zbl 1261.46013)]), the authors are interested in finding metric analogues of this inequality. By this we mean inequalities which, on the one hand, depend only on the metric structure of finite subsets of the space (and thus are well-defined for an arbitrary metric space), and, on the other hand, imply the metric version of the result on poor embeddability of \(\ell_2^n(\ell_1^n)\) into \(L_1\).NEWLINENEWLINEThe authors provide examples showing that the most straightforward and natural metric analogues of the Kwapień-Schütt inequality do not hold in any metric space with at least two elements.NEWLINENEWLINEThe metric analogue suggested in the paper is (the authors use \(\mathbb{F}_2\) for the field with two elements, \(M_n(\mathbb{F}_2)\) for the set of all \(n\times n\) matrices with elements in \(\mathbb{F}_2\), and \(\mathbb{F}_2^n\) for the \(n\)-dimensional Hamming cube): ``Say that a metric space \((X,d_X)\) is a KS space if there exists \(C=C(X)\in (0,\infty)\) such that for every \(n\in 2\mathbb{N}\) and every \(f:M_n(\mathbb{F}_2)\to X\) we have NEWLINE\[NEWLINE \frac{1}{n}\sum_{j=1}^n\sum_{x\in M_n(\mathbb{F}_2)}d_X\Big(f\Big(x+\sum_{k=1}^n e_{jk}\Big),f(x)\Big)\leq \frac{C}{n^n} \sum_{k\in \{1,\dots,n\}^n} \sum_{x\in M_n(\mathbb{F}_2)} d_X\Big(f\Big(x+\sum_{j=1}^n e_{jk_j}\Big),f(x)\Big).\text{''} NEWLINE\]NEWLINE The important difference between this inequality and the Kwapień-Schütt inequality is that the averaging over all permutations \(\pi\in S_n\) is replaced by averaging over all mappings \(\pi:\{1,\dots,n\}\to \{1,\dots,n\}\).NEWLINENEWLINEThe main result of the paper is that \(L_1\) is a KS-space. It is very interesting that in the metric context the proof of this result is simpler than the proof of the original Kwapień-Schütt inequality. The reason is that in the metric case it is natural to apply the result of \textit{I. J. Schoenberg} [Trans. Am. Math. Soc. 44, 522--536 (1938; Zbl 0019.41502; JFM 64.0617.02)] on an isometric embedding of \((L_1,||\cdot||^{1/2})\) into \(\ell_2\), and to reduce the proof to the case of real-valued functions on \(M_n(\mathbb{F}_2)\).NEWLINENEWLINEThis result implies that the (optimal) distortion of bilipschitz embeddings of \(\ell_2^n(\mathbb{F}_2^n)\) into \(L_1\) is of order \(\sqrt{n}\).NEWLINENEWLINEThe authors prove several related results (on \(\ell_q^n(\ell_p^n)\) for \(1\leq p<q\), and on coarse and uniform embeddings, among others) and discuss related open problems.