  In the present paper, we introduce a new concept of convexity which is
generated by a family of endomorphisms of an Abelian group. In Abelian groups
equipped with a translation invariant metric, we define the boundedness, the
norm, the measure of injectivity and the spectral radius of endomorphisms.
Beyond the investigation of their properties, our first main goal is an
extension of the celebrated R\aa dstr\"om Cancellation Theorem. Another result
generalizes the Neumann Invertibility Theorem. Next we define the convexity of
sets with respect to a family of endomorphisms and we describe the
set-theoretical and algebraic structure of the class of such sets. Given a
subset, we also consider the family of endomorphisms that make this subset
convex and we establish the basic properties of this family. Our first main
result establishes conditions which imply midpoint convexity. The next main
result, using our extension of the R\aa dstr\"om Cancellation Theorem, presents
further structural properties of the family of endomorphisms that make a subset
convex.
