Following is a doubt I got when I was reading Csiszar's Annals of Statistics paper "Why least squares and maximum entropy: An axiomatic approach to inference for linear inverse problems". 
He comes up with a set of axioms to justify $\ell\_2$-norm minimisation in $\mathbb{R}^n$. 


He also argues that $\ell\_2$-norm minimization would not be appropriate if the space we consider consists of vectors with positive components,say, $\mathbb{R}\_+^n$. Further he says that there exists sets of the form 


$$L=\{v \in \mathbb{R}\_+^n :Av=b\}$$ 


where $A$ is some $k\times n$ matrix and $u\notin L$ such that $\min\_{v\in L}\|v-u\|\_2$ is not attained in $L$.


I can't think of such an example.


Can someone think of such an example?



Nonexistence of projection
HI Ashok,
 I think $\mathbb{R}^n\_+$ requires strictly positive components. Then it's easy to come up with the situation above, namely consider $L$ to be the part of the line $x+y = 1$ in $\mathbb{R}\_+^2$. And $u = (2,1)$. Then there is no closest point in $L$ to $u$, because it wants to be $(1,0)$, which has nonpositive component. So I guess $n=2$ and $A = (1,1)$ is a $1\times 2$ matrix, and $b=2$. 


