Cone characterization of Grothendieck spaces and Banach spaces containing c0

In this article we study the embeddability of cones in a Banach space X. First we prove that c0 is embeddable in X if and only if its positive cone \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${c_0^+}$$\end{document} is embeddable in X and we study some properties of Banach spaces containing c0 in the light of this result. So, unlike with the positive cone of ℓ1 which is embeddable in any non-reflexive space, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${c_0^+}$$\end{document} has the same behavior as the whole space c0. In the second part of this article we give a characterization of Grothendieck spaces X according to the geometry of cones of X*. By these results we give a partial positive answer to a problem of J.H. Qiu concerning the geometry of cones.