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Let $X$ be a normed space.

Let $K$ be centrally symmetric convex body with $M$ known vertices and with $Vol(K)=V$. We subdivide body $K$ be $M$ equal pieces, $P_i, i=1, \ldots, M$, such that each piece contains only one vertex of $K$.

I am wondering how to find the following minimum: $$ \min_{y\in P_i}\|\sum_{j=1}^nx_jy_jr_j\|^p, $$ where $x_j\in X, y_j \in K$ and $r_j$ is a random variable such that $Prob(r_j=1)=Prob(r_j=-1)=1/2$.

I am not familiar with this kind of objects in math. Maybe its very hard/easy question. Any references or ideas would be very helpful for me.

Edition: The body $K$ is a central slice, formed by the plane $\sum_{i=1}^n r_i=n/2$, perpendicular to the main diagonal of a unit qube $[-1,1]^n$. The reason I would like to find this minimum is because I would like to get a lower bound for the following integral: \begin{align} \int_{K}\|\sum_{j=1}^nx_jy_jr_j(t)\|^pd\mu(y)dt=\sum_{i=1}^M\int_{P_i}\|\sum_{j=1}^nx_jy_jr_j\|^pd\mu(y)dt\\ \geq \sum_{i=1}^M \min_{y\in P_i}\|\sum_{j=1}^nx_jy_jr_j\|^pdt. \end{align} Thank you.

share|improve this question
    
Since $r_j$ are random, should there be expected value $E$ in the formula? –  user31373 Oct 1 '12 at 1:28
    
Yes, this formula is suppose to be under integral over $P_i$. –  user202312 Oct 1 '12 at 3:57
    
So it's not just expectation that is missing, there is also integration with respect to some of the variables? Please edit the formula (the "edit" link is under the question); doing so will improve your odds of getting an answer. –  user31373 Oct 1 '12 at 4:00
    
@LVK: Thank you. I've added the question. –  user202312 Oct 2 '12 at 0:49

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