I am wondering about a natural generalization of theorem 1.4 in the article [*Dvoretzky's theorem — Thirty years later*](https://link.springer.com/article/10.1007/BF01896663) by Milman. My first thought was to look at Milman's paper that he cites for the result (*On a property of functions defined on infinite-dimensional manifolds* ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=309150))), but I have been unable to track it down.


Here is the generalization I'm wondering about:


Let $S^\infty$ be the unit sphere of $\ell\_2$ with real coefficients. Fix a finite integer $k$. Is it true that for any uniformly continuous function $f: (S^\infty)^k \rightarrow \mathbb{R}$ there exists a continuous function $\varphi : [-1,1]^{k \choose 2} \rightarrow \mathbb{R}$ such that for any $\varepsilon >0$ and $n$ there is an $n$-dimensional subspace $E$ of $\ell\_2$ such that for any $x\_0,\dots,x\_{k-1}$ in the unit sphere of $E$, $$|f(x\_0,x\_1,\dots,x\_{k-1})-\varphi\left(r\_{0,1},r\_{0,2},\dots,r\_{0,k-1},r\_{1,2},\dots,r\_{k-2,k-1}\right)| < \varepsilon,$$ where $r\_{i,j}=\left<x\_i,x\_j \right>$ for any $i<j<k$?


Basically, to any given degree of accuracy $\varepsilon$ are there arbitrarily large linear subspaces on which $f(x\_0,\dots,x\_{k-1})$ only depends on the inner products of the $x\_i$ to within $\varepsilon$?



Concentration of measure on finite powers of $S^\infty$
I was able to track down the paper and after Theorem 3, which is the result in question in the $k=2$ case he says:



> 
> Аналогичный теореме 3 факт имеет место и для функции к переменных. Приведем формулировку для трех переменных, поскольку дальнейшие обобщения очевидны.
> 
> 
> 


Which a certain website translates as:



> 
> A fact similar to Theorem 3 holds for the function $k$ of variables. We give the statement for three variables, since further generalizations are obvious.
> 
> 
> 


Which (modulo the misplaced $k$) seems to be saying that Milman does claim the generalization I'm after.


