  We give a Hahn-Banach Characterization for convex-cyclicity. We also obtain
an example of a bounded linear operator $S$ on a Banach space with
$\sigma_{p}(S^*)=\emptyset$ such that $S$ is convex-cyclic, but $S$ is not
weakly hypercyclic and $S^2 $ is not convex-cyclic. This solved two questions
of Rezaei in \cite{Rezaei} when $\sigma_p(S^*)=\varnothing$. %Recently,
Le\'on-Saavedra and Romero de la Rosa \cite{LeRo} provide an example of a
convex-cyclic operator $S$ such that the power $S^n$ fails to be convex-cyclic
with $\sigma _p(S^*)\neq \varnothing$. In fact they solved tree questions posed
by Rezaei in \cite{Rezaei}. Moreover, we prove that $m$-isometries are not
convex-cyclic and that $\varepsilon$-hypercyclic operators are convex-cyclic.
  We also characterize the diagonalizable normal operators that are
convex-cyclic and give a condition on the eigenvalues of an arbitrary operator
for it to be convex-cyclic. We show that certain adjoint multiplication
operators are convex-cyclic and show that some are convex-cyclic but no convex
polynomial of the operator is hypercyclic. Also some adjoint multiplication
operators are convex-cyclic but not 1-weakly hypercyclic.
