Let $P$ be a polytope with $M$ vertices. (The polytope $P$ is the intersection of the hypercube $0≤x _j ≤1$ with the hyperplane $\sum_{j=1}^nx_j=t$, $0\leq t\leq n$). Suppose that the volume of $P$ is $Vol(P)=A$. We subdivide $P$ into $M$ pieces each of the volume $A/M$.
Let $f$ be a function, such that the integral below exist. Then,
$ \int_{P}f(x)ds\geq \sum_{i=1}^{M}\min_{P_i} f(x)|P_i|=\frac{A}{M}\sum_{i=1}^{M}\min_{P_i} f(x). $
Could you please help me to understand the notion of the $\min_{P_i} f(x)$, i.e. what is this and how one can find this minimum?
Thank you for your help.