I have a quesion concerning the likelihood ratio test. In the book the say that the LR statistic $$\lambda=\lambda(X_1,\ldots,X_n)=\frac{\sup_{\Omega_1}L(\theta)}{\sup_{\Omega_0}L(\theta)}$$
Suppose $X\sim N_p(\theta,I)$ and consider testing $H_0: \theta \in \Omega_0$ versus $H_1: \theta \not\in \Omega_0$
Show that the likelihood ratio test statistic $\lambda$ is equivalent to the distance D between $X$ and $\Omega_0$, defined as: $$D=\inf \lbrace ||X-\theta|| : \theta \in \Omega_0 \rbrace$$ (equivalent means there is a one to one increasing relationship between the statistiscs)
Now, a generalized likelihood ratio test will reject $H_0$ if $D>c$. What is is the significance level $\alpha$ of this test if $p=2$ and $\Omega_0=\lbrace \theta : \theta_1\leq 0, \theta_2 \leq 0 \rbrace$ ?
The book does not specify how this obtained ratio is the distance between two statistics. How do I show this? Any help is greatly appreciated :)