The following result is one of several non-linear Banach space theory results known as The Gorelik Principle. I am stating it here in a weaker form than what is in the literature, but the statement here is the full strength required for my purposes.
Suppose that $E,X$ are Banach spaces and $U:E\to X$ is bi-Lipschitz (Lipschitz bijection with Lipschitz inverse). Furthermore, assume $\text{Lip}(U)=1$ and $\text{Lip}(U^{-1})\leqslant d$. Then there exists $\theta>0$ such that for any radius $R>0$ and any subspace $E_0$ of $E$ such that $\dim E/E_0<\infty$, there exists a norm compact subset $K$ of $X$ such that $$RB_X\subset K+U(\theta RB_{E_0}),$$ where $B_Y$ denotes the closed unit ball of $Y$.
Basically this says that finite codimensional subspaces of $E$ are "large enough" that the images of their balls contain the whole ball of $X$, modulo a compact set.
This is useful for arguments regarding asymptotic uniform smoothness and weak$^*$-asymptotic uniform convex. The duality of such properties goes through Young duality, and considers actions of the form $(x^*+\tau x^*_n)(x+\sigma x_n)$, where $x\in X$, $x^*\in X^*$ are fixed based points, $(x^*_n)_{n=1}^\infty$ is a weak$^*$ null sequence (generally nets are required, but let's simplify and assume sequences are good enough for our purposes), and $(x_n)_{n=1}^\infty\subset X$ is a weakly null sequence.
If $E$ has some good asymptotic smoothness property, then one can produce a norm on $X^*$ (which is dual to an equivalent norm on $X$) which has a good weak$^*$ asyptotic convexity property. By the usual duality, you get that the predual norm on $X$ has a good asymptotic smoothness property.
More precisely, one can define the norm $|\cdot|$ on $X^*$ by $$|x^*| = \sup\Bigl\{\frac{|x^*(U(e)-U(e'))|}{\|e-e'\|}:e,e'\in E,e\neq e'\Bigr\}.$$ This is the way this norm has been presented in the literature. One can make the following (potentially not very useful) observation, that really what we are doing here is causing $X$ to be a quotient of the Lipschitz free space $\mathcal{F}(E)$, which causes $X^*$ to be a weak$^*$-continuously embedded subspace of $\text{Lip}_0(E)=\mathcal{F}(E)^*$ via $x^*\mapsto x^*\circ U$. And $|x^*|$ is just the Lipschitz constant of $x^*\circ U$. For this to be completely correct, we need to translate so that $U(0)=0$, where $0$ is both the zero vector and the base point in the Lipschitz free space, but we are fine with translations of $U$.
The Gorelik principle then allows us to press the usual Young duality arguments. Fix $x^*$ with $|x^*|=1$ and fix a weak$^*$-null sequence $(x^*_n)_{n=1}^\infty$ with $\|x^*_n\|=1$ for all $n\in\mathbb{N}$. Pick $e,e'\in E$ such that $|x^*(U(e)-U(e'))|\approx \|e-e'\|$. Again, by some usual translations, we can assume that $e'=-e$ and $U(e)=-U(e')$, so that the Lipschitz constant of $x^*\circ U$ is basically seen on both pairs $0,e$ and $0,-e=e'$. Based on whatever asymptotic smoothness property we have assumed on $E$, there exist $\sigma>0$ and a finite codimensional subspace $E_0$ of $E$ such that for all $f\in \sigma \|e\| B_{E_0}$, $\|e\pm f\|$ will be "good". For example, if we assumed asymptotic uniform flatness, then we can have $\|e\pm f\|\approx \|e\|$. If we assumed $p$-asymptotic uniform smoothness, then we can have $\|e\pm f\|-\|e\|$ to be on the order of $\|e\|\sigma^p$. The goal is then to prove that for all sufficiently large $n$, $|x^*+\tau x^*_n|$ is large (greater than $1+c\tau$ for the asymptotic uniform flatness case, or greater than $1+c\tau^q$ for the $p$-asymptotically uniformly smooth case, where $1/p+1/q=1$ and $c$ is a constant which can depend on $E$, $X$, and $U$, but not on $\tau, \sigma, x, x^*$, or $(x^*_n)_{n=1}^\infty$. For the sake of concreteness, let us focus on the asymptotically uniformly flat case, in which case we do everything for a single, fixed $\sigma$, and all constants are allowed to depend on $\sigma$, because it itself depends only on $E$. In order to complete the usual Young duality, the goal is then to find a sufficiently large $n$, a finite codimensional subspace $E_0$ of $E$, and some $f\in \sigma \|e\|B_{E_0}$ such that
- $x^*_n(U(f))\geqslant D \|e\|$, where $D$ is a constant which can depend on $E,X$, and $U$ but nothing else,
- $x^*_n(U(e))\approx 0$, and
- $x^*(U(f))\approx 0$.
Item 3. is done using the smoothness assumption, and I would like to ignore it. Item 2. is trivial, since $(x^*_n)_{n=1}^\infty$ is weak$^*$-null and $U(e)$ is fixed. It is item 1. that uses the Gorelik principle. The finite codimensional subspace $E_0$ of $E$ comes from $e$ and the asymptotic smoothness assumption. Then the Gorelik Principle says that $\theta^{-1}\|e\|\sigma B_X \subset K+U(\|e\|\sigma B_{E_0})$, where $\theta$ is a constant that depends only on $E,X$, and $U$ and $K$ is norm compact. We can then choose $n$ so large that $x^*_n|_K\approx 0$. We then choose $x=x_0+U(f)\in \theta^{-1}\|e\|B_X$ with $x_0\in K$ and $f\in \|e\|\sigma B_{E_0}$ such that $x^*_n(x)\approx \theta^{-1}\|e\|\sigma$ (recall that $\|x^*_n\|=1$). Since $x^*_n|_K\approx 0$, we can throw away $x_0$ and say that $x^*_n(U(f))\approx \theta^{-1}\|e\|\sigma$, and we get 1.
Let me now say the direction in which I want to take these ideas. Many other nice smoothness properties do not deal with a single base point and weakly null sequence, but with a weakly (or weak$^*$) null tree, which has layers of weakly null sequences. These properties can also be formulated in terms of winning strategies for a player in a two player game. In fact, above, we basically had a two player game where there was only one turn: Once $e$ was fixed, $E_0$ was the choice for a winning strategy. Suppose that for some $L\in\mathbb{N}$, we have a collection $(x^*_G)_{|G|\leqslant L}\subset S_{X^*}$, where the indices run over all non-empty subsets of $\mathbb{N}$ with cardinality not exceeding $L$. Furthermore, assume that all sequences of the form $(x^*_{(m_1, \ldots, m_k, m)})_{m=1}^\infty$ in the collection are weak$^*$-null. The ideal thing would be this: For some $R>0$ (I think the existence of any $R$ for which the following can be carried out is sufficient, so we have some freedom here), the strategy produces some $E_1\subset E$ with $\dim E/E_1<\infty$. By the Gorelik principle, we get some $K_1$ norm compact such that $L^{-1}RB_X\subset K_1 +U(L^{-1}\theta RB_{E_1})$. We choose $n_1$ so large that $x^*_{(n_1)}|_{K_1}\approx 0$. Finally, we choose $f_1\in \theta RB_{E_1}$ such that $x^*_{n_1}(U(f_1/L))\approx L^{-1}\theta R$ (or maybe $cL^{-1}\theta R$ for some uniform constant $c>0$). Once we have $f_1$, the strategy produces some $E_2$, and we choose $K_2$, $n_2$, and $f_2$. We keep going until $n_1, \ldots, n_L$, $f_1, \ldots, f_L$ have been chosen. We want $$\Bigl(\sum_{i=1}^L x^*_{(n_1, \ldots, n_i)}\Bigr)\Bigl(U\bigl(\frac{1}{L}\sum_{i=1}^L f_i\bigr)\Bigr) \geqslant \Delta R,$$ where $\Delta$ is a constant that will depend on $\theta$, $c$, etc., but not on the particular collection $(x^*_G)$.
Here is the problem: The naive attempt to use the Gorelik principle looks identical to the previous argument at the first step of the tree. However, once we get to the second step, in order to apply the principle, we apply it not to the function $U$, but to the function $e\mapsto U(e+f_1/L)-U(f_1/L)$. We can indeed choose $n_2$ so large that $x^*_{n_2}(U(f_1/L))\approx 0$ and we can choose $f_2$ so that $x^*_{n_2}(U(f_2/L+f_1/L))\approx L^{-1}\theta R$ (just like in the previous argument). But the end result of the first step was that $x^*_{n_1}(U(f_1/L))\approx L^{-1}\theta R$, and we have to guarantee that this is not lost when we try to move from $U(f_1/L)$ to $U(f_2/L+f_1/L)$. This is easy to do when $U$ is linear, because then $U(f_2/L+f_1/L)-U(f_1/L)=U(f_2/L)$ can be taken to lie in whatever weak neighborhood of $0$ we want. It is, of course, the exact difficulty of dealing with a non-linear map that we cannot do this when $U$ is not linear. The same problem exists at the general step: We should consider the map $e\mapsto U(e+f_k/L+\ldots+f_1/L)-U(f_k/L+\ldots + f_1/L)$ to choose the next compact set $K_{k+1}$, the next $n_{k+1}$, and the next $f_{k+1}\in \theta RB_{E_{k+1}}$. We need the good property (namely, $x^*_{n_i}(U(f_k/L+\ldots +f_1/L))\approx L^{-1}\theta R$) not to be lost when we move to $U(f_{k+1}/L+\ldots + f_1/L)$.
Is there a way (either within the existing Gorelik principle and its proof or through a stronger version thereof) to proceed with these choices while not losing too much of the functional values obtained at previous steps of the recursion? This looks like an approximate differentiability result to me (the kind which I suspect may be true in a space with good asymptotic smoothness properties). Also, it may be the case that we have to perturb the $f_i$s in the choices above by a small amount in order to land on a sufficiently large set of points of approximate differentiability, or make our choices according to some other additional requirements. But basically we want to know that there's a sufficiently large set of choices for our player in the game.
