Register to reply 
Volume of a convex combination of convex sets ,sort of 
Share this thread: 
#1
Dec1012, 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 BrunnMinkowski theorem, it looks like BrunnMinkowski 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! 


Register to reply 
Related Discussions  
Prove intersection of convex cones is convex.  Calculus & Beyond Homework  8  
Convex Sets  General Math  1  
Prove that the intersection of a number of finite convex sets is also a convex set  Calculus & Beyond Homework  1  
Proving a convex function on an open convex set satisfies some inequalities  Calculus & Beyond Homework  1  
Convex combination and lp's  Calculus & Beyond Homework  0 