Vector Khintchine Inequality Suppose that $X_1,\ldots,X_n$ are fixed vectors in $\mathbb{R}^{d}$ and $\epsilon_1,\ldots,\epsilon_n$ are Rademacher random variables.  Is it the case that there are constants $A_p,B_p$ so that
$$
A_{p}\sqrt{\sum_{i=1}^{n}||X_{i}||_r^{2}}\le \left(\mathbb{E}||\sum_{i=1}^{N}\epsilon_{i}X_{i}||_r^p\right)^{1/p}\le B_{p}\sqrt{\sum_{i=1}^{n}||X_{i}||_r^{2}}
$$
I want a version of the Khintchine inequality that works for $\mathbb{R}^{d}$ equipped with the $r$ norm instead of $\mathbb{C}$ with $|\cdot|$.  If $r$ has to be $2$, that's fine.    
 A: For simplicity, take $p=1$; otherwise use some Jensen-type argument to reduce to the case $p=1$. The sought-for quantity corresponds to the Rademacher complexity of the dual unit-ball $B^*_r := \{w \in \mathbb R^d \mid \|w\|_{r^*} \le 1\}$, where $r^* \in [1,\infty]$ is the harmonic conjugate of $r$, i.e $1/r + 1/r^* = 1$. Upper-bounds of the form you seek have been established in this book (see Lemmas 26.9 and 26.10), for the cases $r \in \{2,\infty\}$.
A: A Rademacher  sequence defines a sequence of i.i.d of Bernoulli random variables taking values $\pm1$ with probability $1/2$ defined on $[0,1],\mathscr{B}([0,1]),m)$ where $m$ is Leebsgue's measure. Thus, we can give a more probabilistic approach.
Consider an i.i.d sequence of Bernoulli random variable $X_n$ with $P[X_n=1]=P[X_n=-1]=\frac12$ and let $X=\sum_na_nX_n$. For any $(a_n:n\in\mathbb{N})\subset\ell_2(\mathbb{N})$ we have that
$$\|\sum^N_{k=n}a_nX_n\|^2_2=\sum_na^2_n$$
Let $N$ be fixed and $S=\sum^N_{n=1}a_nX_n$
and assume that  $\|S\|_2=1$. Then, for any $\lambda>0$
$$\int e^{\lambda S}\,dP=\prod^N_{n=1}\cosh(\lambda a_n)\leq \prod^N_{n=1}e^{\lambda^2a^2_n/2}=e^{\lambda^2/2}$$
Here we have used the fact that $\cosh x\leq e^{x^2/2}$. It then follows that
$$\int e^{\lambda|S|}dP\leq \int e^{\lambda S}dP+ \int e^{-\lambda S_N}dP\leq 2e^{\lambda^2/2}$$
Here we use the fact that $(-X_n)\stackrel{d}{=}(X_n)$. By Markov-Chebyshev's inequality
$$
P[|S|>\lambda]\leq e^{-\lambda^2}E\big[e^{\lambda|S|}\big]\leq 2e^{-\lambda^2/2}$$
For $p\geq2$
\begin{align}
\int |S|^p\,dP&=p\int^\infty_0 \lambda^{p-1}P[|S|>\lambda]\,d\lambda\\
&\leq 2p\int^\infty_0 \lambda^{p-1}e^{-\lambda^2/2}\,d\lambda\\
&=2p\int\int^\infty_0(2t)^{p-2}e^{-t}\,dt\\
&=2^{p/2+1}\Gamma\big(p/2 +1\big)=:K^p_p
\end{align}
Here we use the change of variables $t=\lambda^2/2$.  All this means that
for any finite linear combination $X=\sum^N_{n=1}a_nX_n$,
$$\|X\|_p\leq K_p\|X\|_2$$
where $K_p=\Big(2^{p/2+1}\Gamma\big(p/2 +1\big)\Big)^{1/p}$ if $p>2$ and $K_p=1$ for $0<p<2$.
For $0<p<2$, application of Hölder's inequality yields
\begin{align}
\|X\|^2_2&=\int|X|^2\,dP=\int|S_N|^{p/2}|X|^{2-\tfrac{p}{2}}\,dP\\
&\leq \|X\|^{p/2}_p\|X\|^{2-\tfrac{p}{2}}_{4-p}\\
&\leq \|X\|^{p/2}_pK^{2-\tfrac{p}{2}}_{4-p}\|X\|^{2-\tfrac{p}{2}}_2
\end{align}
Consequently
$$\|X\|_2\leq k_p\|X\|_p$$
where $k_p=1$ if $p\geq 2$ and $k_p=\big(K_{4-p}\big)^{\tfrac{4}{p}-1}$ for $0<p<2$.
