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 $$(1)\qquad E f\Big(\Big\|\sum_{i=1}^n\varepsilon_i x_i\Big\|\Big) \le E f\Big(\sum_{i=1}^n\varepsilon_i \|x_i\|\Big) $$ 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.