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

1. Dec 10, 2012

### hwangii

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

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!