  We introduce and study a new class of $\eps$-convex bodies (extending the
class of convex bodies) in metric and normed linear spaces. We analyze
relations between characteristic properties of convex bodies, demonstrate how
$\eps$-convex bodies connect with some classical results of Convex Geometry, as
Helly theorem, and find applications to geometric tomography. We introduce the
notion of a circular projection and investigate the problem of determination of
$\eps$-convex bodies by their projection-type images. The results generalize
corresponding stability theorems by H. Groemer.
