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Volume of a convex combination of convex sets ,sort of

by hwangii
Tags: combination, convex, sets, sort, volume
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hwangii
#1
Dec10-12, 01:50 PM
P: 4
Hi all,
I hope someone can tell me whether this is true or not:

Let [itex]A_{i},i=\{1,...,m\}[/itex] be [itex]m \times n[/itex] matrices, and let
[itex]H_{i}=\{x\in \mathbb{R}^{n}:A_{i}x\geq 0\},i=\{1,...,m\}.[/itex] Also let a probability measure [itex]\mu[/itex] be given.
Define
[itex]H(\lambda)=\{x\in\mathbb{R}^{n}:\sum_{i=1}^{m} \lambda_{i} A_{i} x\geq 0\}[/itex] where [itex]\lambda=(\lambda_{1},...,\lambda_{m}) \in \mathbb{R}^{m}[/itex] and [itex]\forall i\in\{1,...,m\},\lambda_{i}\geq 0,\sum_{i=1}^{m}\lambda_{i}=1.[/itex]
Then is the following true?
[itex]\mu(H(\lambda)) \geq min_{i \in \{1,...,m\}} \mu(H_{i})[/itex]
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 [itex]H(\lambda)[/itex] is not a linear combination of [itex]H_{i},i=\{1,...,m\}[/itex], so I don't know if there is some version of the theorem that is applicable to my question.
Thanks!
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