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.