How can I estimate the volume of the region inside an open bounded set in dimension n at distance less than 1/n from its boundary?

Question

How can I estimate the volume of the region inside an open bounded set in dimension $n$ at distance less than $1/n$ from its boundary?

Answer 1

Suppose that your region $R$ is contained in an open set $U$. Then you can rephrase the condition "the distance between $R$ and the boundary of $U$ is less than $1/n$" as $$ U \supseteq R \supseteq \frac{n-1}{n} U $$ which means that $$ \operatorname{vol}(U) \geq \operatorname{vol}(R) \geq \left(\frac{n-1}{n}\right)^n \operatorname{vol}(U). $$