"Measuring" how far is one Banach space from being surjectively isometric to another

Question

Bonjour/bonsoir à toutes et à tous.

Assume that $\mathbf{V} \equiv (V, \|\cdot\|_V)$ and $\mathbf{W} \equiv (W, \|\cdot\|_W)$ are Banach spaces (over the real or complex field).

Question 1. What are some appropriate indices you might use to "measure" how far is $\mathbf{V}$ from being i) surjectively isometric (see note N1) or ii) isometrically isomorphic to $\mathbf{W}$ (see note N2)?

I am conscious that the question may sound a little vague, so take the Banach-Mazur distance as a practical example of what I (am trying to) mean.

Added later. After an answer by Bill Johnson (see below), I'm adding here that another index (in the sense of Question 1) is given, for the non-linear case, by the Lipschitz distance (or Lipschitz distorsion). This is known to be the same as the Banach-Mazur distance so far as $\mathbf{V}$ and $\mathbf{W}$ are (isomorphic and) finite-dimensional. Yet, as still pointed out by BJ, the same question, when raised in the infinite-dimensional setting with regard to the separable case, is an open problem to date. A further possibility, when $\dim(V) = \dim(W) < \infty$, is given by the so-called weak Banach-Mazur distance (see my comment to Bill's first answer for a reference).

Question 2. Could you provide some concrete examples illustrating why, depending on the case, the one index should be preferred to the others (if any)?

My apologies in advance if the question has been already asked.

Notes. (N1) Following a comment by Yemon Choi, I emphasize that, unless differently stated, I am using the term isometry to refer to both linear and non-linear isometries. (N2) Of course, in the real case, there is no true need to distinguish between conditions i) and ii) in the statement of Question 1 (by the Mazur-Ulam theorem).

Answer 1

For (i) the usual thing is to take the Lipschitz analogue of the Banach-Mazur distance; namely, the infimum over injective and surjective maps $T$ from $V$ to $W$ of the Lipschitz constant of $T$ times the Lipschitz constant of $T^{-1}$. Whether this is equivalent to the Banach-Mazur distance for separable Banach spaces is a well known open problem. See the book by Benyamini and Lindenstrauss.

Answer 2

Rather than talk about the weak distance and distance, it is better to discuss the weak factorization constant and the factorization constant of an operator $u$ through an operator $T$. The factorization constant of $u: X\to Y$ through $T:Z\to W$, $\gamma_T(u)$, is the infimum of $\|\alpha\|\cdot \|\beta\|$ over all $\alpha:X\to Z$ and $\beta:W\to Y$ for which $\beta T \alpha =u$. This measurement of the size of $u$ is generally not a norm, but you can convexify it to get the weak factorization constant, $\hat{\gamma}_T(u)$, of $u$ through $T$, which is defined to be the infimum of $\sum_i \gamma_T(u_i)$ s.t. $u=\sum_i u_i$. The (weak) factorization constant of $u$ through a space $Z$ is just the (weak) factorization constant of $u$ through $I_Z$. Obviously you can write down the distance and weak distance in terms of factorization and weak factorization constants.

One classical situation in which these parameters differ a lot is in my Studia Math. 89 (1988), 79--103 paper with Figiel and Schechtman. Let $u$ be the basis to basis mapping from $\ell_2^n$ to the first $n$ Rademacher functions in $L_1$. The factorization constant of this operator through $\ell_1^{Cn}$ is large for any fixed $C$, but the weak factorization constant through $\ell_1^n$ is bounded independently of $n$. That is, you cannot well factor this embedding of $\ell_2^n$ through a low dimensional $L_1$ space, but you can well weakly factor it through $\ell_1^n$ (in fact, any operator from $\ell_2^n$ into $L_1$ well weakly factors through $\ell_1^n$; see Proposition 5.5 of the paper I mentioned above). We also show that if you want to well factor this Rademacher embedding $u$ through $\ell_1^k$, then $k$ must be at least exponential in $n$.