# Volume of a convex combination of convex sets ,sort of

by hwangii
Tags: combination, convex, sets, sort, volume
 P: 4 Hi all, I hope someone can tell me whether this is true or not: Let $A_{i},i=\{1,...,m\}$ be $m \times n$ matrices, and let $H_{i}=\{x\in \mathbb{R}^{n}:A_{i}x\geq 0\},i=\{1,...,m\}.$ Also let a probability measure $\mu$ be given. Define $H(\lambda)=\{x\in\mathbb{R}^{n}:\sum_{i=1}^{m} \lambda_{i} A_{i} x\geq 0\}$ where $\lambda=(\lambda_{1},...,\lambda_{m}) \in \mathbb{R}^{m}$ and $\forall i\in\{1,...,m\},\lambda_{i}\geq 0,\sum_{i=1}^{m}\lambda_{i}=1.$ Then is the following true? $\mu(H(\lambda)) \geq min_{i \in \{1,...,m\}} \mu(H_{i})$ My guess is that this has something to do with Brunn-Minkowski theorem, it looks like Brunn-Minkowski theorem is for linear combinations of convex sets, but my $H(\lambda)$ is not a linear combination of $H_{i},i=\{1,...,m\}$, so I don't know if there is some version of the theorem that is applicable to my question. Thanks!