application of some inequalities in optimization theory

Question

I am studding the Theory of Optimization. And it turns out that some classical inequalities (especially like Khinchine, or non-commutative Khinchine inequaliry or Kahane's inequality) are 'very key' in this subject. I am wondering, would it be interesting to consider in the Optimizatio Theory these inequlities with some extra assumptions on the 'variables'?

For example, the classical Khinchine inequality states that for vector $x=(x_1, \ldots, x_{2m})\in R^{2m}$, for $p\geq 2$, and for independent Rademacher random variables $r_1, \ldots, r_{2m}$ ($P(r_i=1)=P(r_i=-1)=\frac{1}{2}$) one has $ \mathbb{E}\left(\left|\sum_{i=1}^{2m}r_ix_i\right|^p\right)^{1/p}\leqslant C\sqrt{p}\, \left\|x\right\|_2, $

Would it be interesting to consider such inequlity under the various conditions, i.e. conditions on the vector $x$ and conditions on the sequence of Rademacher random variables?

Thank you.