Let $x_1,\dots,x_n$ be vectors in a Euclidean space $H$. Let $\varepsilon_1,\dots,\varepsilon_n$ be independent Rademacher random variables, so that $P(\varepsilon_i=\pm1)=1/2$ for all $i$. 
Does the inequality 
$$(1)\qquad E f\Bigl(\Bigl\|\sum_{i=1}^n\varepsilon_i x_i\Bigr\|\Bigr)
\le E f\Bigl(\sum_{i=1}^n\varepsilon_i \|x_i\|\Bigr)
$$
always hold if $f(x)=f_p(x):=|x|^p$ for some real $p\ge2$ and all real $x$? For $f=f_p$ with $p\ge3$, this inequality is due to Utev [1].   

As noted in [2], it is enough to prove $(1)$ for $f=g_a$, where $a\ge0$ and $g_a(x):=\max(0,|x|-a)^2$ for all real $x$. It is also shown in [2] (where more context can be found) that one may assume without loss of generality that the dimension of the Euclidean space $H$ is no greater than $\left\lfloor(\sqrt{1+8n}-1)/2\right\rfloor\sim\sqrt{2n}$.

**References**

[1] Utev, S.A., Extremal problems in moment inequalities. (Russian) *Limit theorems of probability theory*, 56--75, 175,
Trudy Inst. Mat., 5, *``Nauka'' Sibirsk. Otdel., Novosibirsk*, 1985; MathSciNet Review MR0821753. 

[2] Pinelis, I., Dimensionality reduction in extremal problems for moments of linear combinations of vectors with random coefficients. *Stochastic inequalities and applications*, 169--185, Progr. Probab., 56, *Birkhäuser, Basel*, 2003; MathSciNet Review MR2073433.