Here is a proof for the case of a convex polytope.
If $P\subset \Bbb{R}^d$ is a compact convex polytope containing the origin $O$ in its interior then it can be broken up into cones over its facets and it suffices to verify the inequality for each of these (polyhedral) cones $C$ individually. Let $F$ be a facet of $C$ with area $A$. The hyperplane supporting $F$ separates $\Bbb{R}^d$ into two half-spaces, the inner one containing $O$ and the outer one. Then $B_{\epsilon}(F)\cap C$, the "inner" $\epsilon$-neigborhood of $F$, is contained in the inner cylinder with height $\epsilon$ based on $F$ and $B_{\epsilon}(F)\cap \overline{C}$, the "outer" $\epsilon$-neighborhood of $F$, contains the outer cylinder with height $\epsilon$ based on $F$. Therefore,
(the inner volume) $\leq \epsilon A\leq $ (the outer volume).