We extend the notion of convex-cyclicity for matrices to that of convex-cyclicity for abelian semigroups of matrices on \({\mathbb {K}}^{n}\) . We say that an abelian semigroup G of matrices on \({\mathbb {K}}^{n}\) is convex-cyclic if there exists \(x\in {\mathbb {K}}^{n}\) such that the convex hull of the orbit G(x) of x under G is dense in \({\mathbb {K}}^{n}\) . We provide an effective method for checking that a given abelian semigroup is convex-cyclic. In particular we give a spectral characterization of convex-cyclicity for finitely generated abelian semigroups of diagonalizable matrices. We also obtain an example of a convex-cyclic abelian semigroup which does not contain a convex-cyclic matrix. We prove that there exists an abelian semigroup of matrices on \({\mathbb {K}}^{n}\) which is somewhere convex-cyclic but not convex-cyclic. This solves two questions of Rezaei in (Linear Algebra Appl 438:4190–4203, 2013) and Feldman and McGuire in (Oper Matrices 12(2):465–492, 2017) respectively. On the other hand, we provide the link between \(\varepsilon \) -hypercyclicity and convex-cyclicity for abelian semigroups of matrices on \({\mathbb {K}}^{n}\) for every \(0<\varepsilon <1\) .
