Before I ask my question I will provide a brief introduction.

I came across the notion of Rademacher type while reading Assaf Naor's article An introduction to the Ribe program, which can be found here http://arxiv.org/pdf/1205.5993v1.pdf.

A Banach space $(X,||\cdot||_X)$ is said to have Rademacher type $p \geqslant 1$ if there exists a constant $T \in (0,\infty)$ so that for every $n \in \mathbb{N}$ and for every $x_1,\ldots,x_n \in X$ we have \begin{equation} \frac{1}{2^n}\sum_{\varepsilon_1,\ldots,\varepsilon_n \in \{-1,+1\}} \left|\left|\sum_{i=1}^n \varepsilon_i x_i \right|\right|_X \leqslant T\left(\sum_{i=1}^n ||x_i||_X^p\right)^{1/p}.\end{equation}

In the article, Naor claims that $p \leqslant 2$ is immediate by considering the case when $x_1,\ldots,x_n$ are colinear. I can prove this using probabilistic techniques as follows.

Let $(X,||\cdot||_X)$ satisfy the above inequality and consider the colinear vectors $x_1,\ldots,x_n$, where each $x_i$, $i = 1,\ldots,n$, is simply a copy of a fixed vector $x \in X$. By factoring we can cancel $||x||_X$ out from both sides, leaving \begin{equation} \frac{1}{2^n}\sum_{\varepsilon_1,\ldots,\varepsilon_n \in \{-1,+1\}} \left|\sum_{i=1}^n \varepsilon_i\right|\leqslant Tn^{1/p}.\end{equation} Now we interpret the left side as the expected distance from the origin after $n$ steps of the simple random walk in $\mathbb{Z}$. It is well known that the drift of this walk, $\mathbb{E}[|S_n|]$, satisfies $$cn^{1/2} \leqslant \mathbb{E}[S_n] \leqslant Cn^{1/2}$$ for some constants $c,C > 0$. Consequently it follows that $p \leqslant 2$.

My question is as follows: is there a more functional analytic way to conclude that the definition of Rademacher type does not make sense for $p > 2$? Since the notion of Rademacher type is a strengthening of the randomized triangle inequality, I am not too surprised that probabilistic techniques pop up, yet I am curious for an alternate approach.

Please let me know if this question is not 'research worthy' enough to be asked here, but belongs elsewhere, such as on StackExchange.



If you you use the second moment instead of the first moment to define type you can use the parallelogram law (or orthogonality of symmetric Bernoulli random variables) to get:

\begin{equation} \frac{1}{2^n}\sum_{\varepsilon_1,\ldots,\varepsilon_n \in \{-1,+1\}} \left|\sum_{i=1}^n \varepsilon_i\right|^2 = n.\end{equation}

This shows that the real line and hence every normed space of dimension at least one cannot have type larger than $p$ when you use the second moment to define type. To come back to the first moment, you need to use Khintchine's inequality, for which there are proofs that do not use probability theory (e.g., prove the inequality for $p=4$ by multiplying things out and use extrapolation to deduce the inequality for $p=1$).

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