Let $C(\omega^\omega+)$ denote the Banach space of continuous, scalar-valued functions defined on $\omega^\omega+=[0,\omega^\omega]$. Suppose that $X$ is a Banach space and $U:C(\omega^\omega+)\to X$ is a Lipschitz isomorphism (that is, a Lipschitz bijection whose inverse is also Lipschitz).
Let $T_n=\{(k_i)_{i=1}^j:j\leqslant n, k_i\in \mathbb{N}\}$. That is, $T_n$ is the set of all sequences of natural numbers the length of which lies in $[1,n]$. We say $(x^*_t)_{t\in T_n}\subset X^*$ is \emph{weak}$^*$-\emph{null} if
- $(x_{(k)})_{k=1}^\infty$ is weak$^*$-convergent to $0$, and
- for each $(k_1, \ldots, k_j)\in T_{n-1}$, $(x_{(k_1, \ldots, k_j, k)})_{k=1}^\infty$ is weak$^*$-convergent to $0$.
Let $\text{codim}(C(\omega^\omega+))$ denote the set of closed, finite co-dimensional subspaces of $C(\omega^\omega+)$. For $r,\vartheta>0$, $n\in\mathbb{N}$, and a weak$^*$-null $(x^*_t)_{t\in T_n}\subset S_{X^*}$ (the unit sphere of $X^*$), we consider a two-player game. Player I chooses $E_1\in \text{codim}(C(\omega^\omega+))$ and $l_1\in \mathbb{N}$. Player II chooses $e_1\in rB_{E_1}$ and $l_1<k_1\in \mathbb{N}$. Player I chooses $E_2\in \text{codim}(C(\omega^\omega+))$ and $l_2\in \mathbb{N}$ and and Player II chooses $e_2\in rB_{E_2}$ and $l_2<k_2\in \mathbb{N}$. Play continues in this way until $e_1, \ldots, e_n$ and $k_1, \ldots, k_n$ are chosen. Player I wins if $$\Bigl|\Bigl\langle x^*_{(k_1)}+x^*_{(k_1,k_2)}+\ldots + x^*_{(k_1, \ldots, k_n)},U\Bigl(\frac{1}{n}\sum_{i=1}^n e_i\Bigr) \Bigr\rangle\Bigr|<r\vartheta,$$ and Player II wins otherwise.
I would like to know whether there exists $\vartheta>0$ (depending on $X$ and $U$ but nothing else) such that for any $r>0$ and $n\in \mathbb{N}$, Player II has a winning strategy in this game.
It is easy to see that the statement holds when $U$ is linear. This is because "far out" $x^*$ ($x^*$ in a prescribed, sufficiently small weak$^*$-neighborhood of $0$) can be well-normed (up to a factor of $1/2-\varepsilon$) by vectors in a given $E_i\in \text{codim}(C(\omega^\omega+))$. We can use this fact plus linearity to deduce that $|\langle x^*_{(k_1, \ldots, k_j)}, U(e_m)\rangle|\approx 0$ for $j\neq m$, and $\text{Re }\langle x^*_{(k_1, \ldots, k_n)},U(e_j)\rangle > r/3$. So Player II will have a winning strategy with $\vartheta=1/4$. Actually any $\vartheta\in (0,1/2)$ works. Also, in the linear case, by homogeneity we can ignore the $r$.
The result also holds if we only care about $n=1$. This is because of one of several results called The Gorelik Principle, as used here. In Proposition 2.7 of the linked paper, they prove that for any Banach spaces $E,X$ and any Lipschitz isomorphism $U:E\to X$, there exists $\Theta>0$ such that for any finite co-dimensional subspace $E_1$ of $E$ and any $r>0$, there exists a norm compact subset $K$ of $X$ such that for any $r>0$, $$\Theta r B_X \subset K + U(r B_{E_1}).$$ Here $\theta$ is something like $\frac{1}{2\text{Lip}(U^{-1})}$. Then for any weak$^*$-null sequence $(x^*_k)_{k=1}^\infty$, there exists $k$ so large that $x^*_k|_K\approx 0$. We choose $x\in \Theta rB_X$ such that $\langle x^*_k,x\rangle\approx \Theta r$, decompose $x=x'+U(e)$ for $e\in rB_{E_1}$, and note that $\langle x^*_k,U(e)\rangle\approx \Theta r$ still, since $|\langle x^*_k,x'\rangle|\approx 0$.
The naive approach is to continue the argument from the linear case, using the Gorelik Principle at each step. This allows us to choose $e_1, \ldots, e_n$, $k_1, \ldots, k_n$ such that $$\Bigl\langle x^*_{k_j}, U\Bigl(\frac{1}{n}\sum_{i=1}^j e_i\Bigr) - U\Bigl(\frac{1}{n}\sum_{i=1}^{j-1} e_i\Bigr)\Bigr\rangle\approx \Theta r/n$$ and $$\Bigl|\Bigl\langle x^*_{k_j}, U\Bigl(\frac{1}{n}\sum_{i=1}^{j-1} e_i\Bigr)\Bigr\rangle\Bigr|\approx 0,$$ so that $$\Bigl\langle x^*_{k_j}, U\Bigl(\frac{1}{n}\sum_{i=1}^j e_i\Bigr) \Bigr\rangle\approx \Theta r/n.$$ But since $U$ is non-linear, and therefore need not take weakly null nets to weakly null nets, this approach does not enable us to control $$\Bigl \langle x^*_{k_j}, U\Bigl(\frac{1}{n}\sum_{i=1}^m e_i\Bigr)\Bigr\rangle$$ for $m=j+1, j+2, \ldots, n$ in terms of $$\Bigl \langle x^*_{k_j}, U\Bigl(\frac{1}{n}\sum_{i=1}^j e_i\Bigr)\Bigr\rangle ,$$ where we obtained the desired value.
Can we improve on the naive approach?
