Let $\mu \in \mathbb R^n$ and let $\Sigma$ be a positive-definite matrix of order $n \ge 2$. Fix $t \ge 0$ and define $\alpha(\mu,\Sigma,t) > 0$ by

$$ \alpha(\mu,\Sigma,t) := \sup_{\|w\| = 1}\frac{1}{\|w\|_\Sigma}\varphi\left(\frac{w^\top \mu - t}{\|w\|_\Sigma}\right), $$

where $\|w\|_\Sigma := \sqrt{w^\top \Sigma w}$ and $\varphi$ is the Gaussian pdf. Also define $\beta(\mu,\Sigma,t) \ge 0$ by $$ \beta(\mu,\Sigma,t) := \inf_{\|z\| \le t}\|z-\mu\|_{\Sigma^{-1}}. $$

Question.Is there any functional relationship between $\alpha(\mu,\Sigma,t)$ and $\beta(\mu,\Sigma,t)$ ?

## Example: isotropic case

Suppose $\Sigma = I_n$, the identity matrix. Then

$\alpha(\mu,\Sigma,t) = \varphi(r_\star)$, where $$ \begin{split} -r_\star := \inf_{\|w\| = 1}|w^\top \mu - t| &= \inf_{\|w\| = 1}\sup_{s \in \{\pm 1\}}s(w^\top \mu-t) = \sup_{s \in \{\pm 1\}}-st+\inf_{\|w\| = 1}sw^\top \mu\\ & = \sup_{s \in \{\pm 1\}}-st-\|\mu\|=t-\|\mu\|, \end{split} $$ if $\|\mu\| \ge t$, and $r_\star = 0$ otherwise. That is, $r_\star = (\|\mu\|-t)_+$.

On the other hand, one computes $$ \begin{split} \inf_{\|z\| \le 1}\|z-\mu\| &= \begin{cases} 0,&\mbox{ if }\|\mu\| \le t,\\ \|t\mu/\|\mu\|-\mu\| = |t-\mu| = \|\mu\|-t,&\mbox{ else} \end{cases}\\ &=(\|\mu\|-t)_+ = r_\star. \end{split} $$

We conclude

$\alpha(\mu,I_n,t) = \varphi(\beta(\mu,I_n,t))$.