Is the following set compact, is it convex and what is the convex hull?
- $V = \{(x_1, x_2,...,x_n) \in \mathbb{R}^n :\frac{1}{1 + i} \leq x_i \leq \frac{1}{i}, i=1,2,...,n\}$
My thoughts:
I was thinking that it should be compact, as the series is finite and thus there should exist a finite sub covering of the open cover.
When it comes to the convexity I don't know how to apply the general formula $\lambda x + (1-\lambda)y$ to those equations above.
In case the set would be convex, the convex hull were the set itself, right? But in case it is not convex, it should be a set that contains all convex combinations. Is there any form of calculation for the convex hull? Thanks a lot in advance for your help!
Yes. It is compact, just as you think.
In fact $V=[\frac12,1] \times [\frac13, \frac12]\times \cdots \times [\frac{1}{n+1},\frac1n]$. By using a famuous theorem, (sorry, I forget its name), that any products of compact spaces is still compact, we conclude that $V$ is compact.
Is it convex? I am not sure. But we see $[\frac12,1],[\frac13, \frac12], \cdots [\frac{1}{n+1},\frac1n]$ are all convex.
If the following claim is true, we also can conclude that $V$ is convex.
Any products of finite convex spaces is still convex
Hope it can be helpful for you.