How to get the upper bound of $\sup_{z\in\partial{B_0}}\exp\left[-\frac{d(z,y)^2}{C}\right]$

Question

How to estimate the maximal of the following formula:

$$\sup_{z\in\partial{B_0}}\exp\left[-\frac{d(z,y)^2}{C}\right]\leq ?$$

where $B_0:=B(x_0, R)$ and $z, y\in B(x_0, R)$ and $C$ is some constant.

Answer 1

For $C>0$ (which I assume is the case) the function

$$ \exp{\left (-\frac{d^2}{C} \right)} $$

is a decreasing function of $d^2$. Hence its supremum is the infimum of $d^2$, which is

$$ L^2:=\left [ \inf_{z\in \partial B_0} d(z,y) \right ]^2 $$

Hence $L$ is equal to the distance of $y$ from the surface of the sfere of radius $R$.

Ultimately

$$\sup_{z\in\partial{B_0}}\exp\left[-\frac{d(z,y)^2}{C}\right]\leq \exp{\left (-\frac{L^2}{C} \right)} \le 1 $$