# application of some inequalities in optimization theory

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.

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Again, you can get vertical bars of appropriate sizes by using \left and \right in front of them. The same with parentheses. –  joriki Aug 7 '12 at 2:28
Cross-posted on cstheory.SE –  Sasha Aug 7 '12 at 3:35
@Sasha: Yes, I've seen this post. And I have slitely different condition... Plus, I would like to find some application in math... –  Michael Aug 7 '12 at 4:19
In particular, I am interested in the optimization theory. I've changed my question. –  Michael Aug 7 '12 at 4:48
@Michael: How do you mean, "I've seen this post"? Your original question was word by word identical with it. Are you implying that you neither posted both questions nor copied the other question here? (That's what "I've seen this post" would usually be taken to imply.) If so, how did the identical wording come about? –  joriki Aug 7 '12 at 11:25