Different forms of metric characterizations of classes of Banach spaces (Q2862653)

The main topic of this article is the study of metric characterizations of classes of Banach spaces in terms of test-spaces. For metric spaces \(X\) and \(Y\), let NEWLINE\[NEWLINE c_Y (X) := \inf\{\text{Lip}(f )\text{Lip}(f ^{-1} )\mid f : X \to Y \text{ injective}\}NEWLINE\]NEWLINE denote the \(Y\)-distortion of \(X\). A sequence of metric spaces \((T_n )_{n\in \mathbb N}\) is a sequence of test-spaces for a class \(\mathcal P\) of Banach spaces if the following two conditions are equivalent:NEWLINENEWLINE(1) \(Y \in P\).NEWLINENEWLINE(2) \(\sup_{n\in \mathbb N} c_Y (T_n ) < \infty\).NEWLINENEWLINEIf \(\sup_{n\in \mathbb N} c_Y (T_n ) < \infty\), then we say that the sequence \((T_n )_{n\in \mathbb N}\) equi-Lipschitzly embeds into \(Y\). The main result of the article says that if a class of Banach spaces admits a sequence of test-spaces formed by finite-dimensional Banach spaces, it also admits a sequence of test-spaces formed by finite unweighted graphs with maximum degree 3. An unweighted graph is always equipped with its canonical shortest path metric. More precisely, the author proves:NEWLINENEWLINE Theorem. Let \(Y\) be a Banach space. Let \((F_n )_{n\in \mathbb N}\) be a sequence of finite-dimensional Banach spaces with \(\sup_{n\in \mathbb N} \dim(F_n ) = \infty\). Then there exists a sequence of finite unweighted graphs \((G_k )_{k\in \mathbb N}\), with maximum degree 3, such that NEWLINE\[NEWLINE\sup_{n\in \mathbb N} c_Y (F_n ) < \infty \text{ if and only if } \sup_{k\in \mathbb N} c_Y (G_k ) < \infty.NEWLINE\]NEWLINE The proof of the theorem goes along the following lines. The author starts by proving a relaxation of the theorem where the maximum degree of the graphs is not required to be bounded. He shows how one can construct a graph \(G(F, \delta, r)\) out of a \(\delta\)-net of the ball of radius \(r\) centered at the origin of some finite-dimensional Banach space \(F\) that admits a bi-Lipschitz embedding with distortion at most 3 into \(F\). For a sequence \((F_n )_{n\in \mathbb N}\) of finite-dimensional Banach spaces with \(\sup_{n\in \mathbb N} \dim(F_n ) = \infty\), such a construction will certainly produce a sequence of graphs \((\tilde{G}_k )_{k\in \mathbb N}\), with inherently unbounded degrees though, that equi-Lipschitzly embeds into \(Y\) as soon as the sequence \((F_n )_{n\in \mathbb N}\) does. To make sure that the equi-Lipschitz embeddability of the sequence \((F_n )_{k\in \mathbb N}\) is sufficient to imply the equi-Lipschitz embeddability of the sequence \((F_n )_{n\in \mathbb N}\), the author chooses finer and finer nets of larger and larger balls, namely, \((\tilde{G}_k )_{k\in \mathbb N}\) is an enumeration of the double-indexed sequence \(\{G(F_n ,\frac 1m , m)\}_{m\in \mathbb N,n\in \mathbb N}\). The difficult task is to guarantee that the previous construction can be modified to obtain a sequence of graphs with degrees bounded above by 3. The necessary modification requires two main ingredients. One of them is the simple observation that a finite graph always admits a bi-Lipschitz embedding, say with distortion 2, into a finite graph with maximum degree 3. The idea is to replace each vertex \(v\) of the original graph by a cluster containing at least as many points as incident edges (for instance replacing a vertex of degree, say 5, by a path or a cycle on 5 vertices will work), and connect the clusters by long paths according to the original graph structure such that a new vertex in a cluster must belong to only one long path. It is immediate that the new graph obtained has maximum degree 3, and it is not hard to verify that it contains a bi-Lipschitz copy of the original graph with distortion as close to 1 as wanted if the long paths are chosen long enough. The other ingredient involves the thickening of an unweighted graph. One can thicken a graph \(G\) by replacing each edge with an isometric copy of the segment \([0, 1]\). The distance between points in the thickening \(T G\) of \(G\) is the length of the shortest curve joining the points. Using tedious volumic arguments, the author shows that, if \(F\) is a Banach space of dimension at least 3, then the thickening \(T G(F, \delta, r)\) of the graph \(G(F, \delta, r)\) above admits a bi-Lipschitz embedding into \(F\) whose distortion is independent of \(F , \delta\), and \(r\).NEWLINENEWLINEThe article also contains a proof of the observation that, if a finite simple connected graph \(G\) admits an isometric embedding into a strictly convex Banach space, then \(G\) is isomorphic to either a complete graph or a path.