I am sure it IS in Gruber's book - what you want is Steiner's formula on the volume of parallel surfaces (or tubular neighborhoods of the boundary, if you prefer) of a convex body. These have the form:

$$V_r = \sum_{i=0}^d Q_i r^i,$$ where the $Q_i$ are the Quermassintegrals, which are positive, $r$ is the distance along the outward normal and $d$ the dimension. $Q_0$ is the area of the boundary. If $r$ (which corresponds to your $\epsilon$ is small enough), then the $i=1$ term dominates the perturbation part ($Q_0$ is what it is), and so your desired result follows.

This has been generalized by Weyl to Riemannian manifolds, and there is a whole (quite good) book on the subject - Tubes, by Alfred Gray.

I am sure it IS in Gruber's book - what you want is Steiner's formula on the volume of parallel bodies (or tubular neighborhoods, if you prefer) of a convex body. These have the form:

$$V_r = \sum_{i=0}^d Q_i r^i,$$ where the $Q_i$ are the Quermassintegrals, which are positive, $r$ is the distance along the outward normal and $d$ the dimension. $Q_1$ is the area of the boundary. Consider the epsilon-parallel body $B_\epsilon$ inside $B$ Then, the volume of the inside neighborhood is $$ V(\epsilon) = \sum_{i=1}^d Q_i(B_\epsilon) \epsilon^i,$$ while the volume of the outside $|epsilon$-neighborhood is $$V_{2\epsilon} - V_\epsilon.$$ Notice that all the terms in the difference are positive.

This has been generalized by Weyl to Riemannian manifolds, and there is a whole (quite good) book on the subject - Tubes, by Alfred Gray.