Properties associated with the epigraph of the $$l_1$$l1 norm function of projection onto the nonnegative orthant

This paper studies some properties associated with a closed convex cone $$\mathcal {K}_{1+}$$K1+, which is defined as the epigraph of the $$l_1$$l1 norm function of the metric projection onto the nonnegative orthant. Specifically, this paper presents some properties on variational geometry of $$\mathcal {K}_{1+}$$K1+ such as the dual cone, the tangent cone, the normal cone, the critical cone and its convex hull, and others; as well as the differential properties of the metric projection onto $$\mathcal {K}_{1+}$$K1+ including the directional derivative, the B-subdifferential, and the Clarke’s generalized Jacobian. These results presented in this paper lay a foundation for future work on sensitivity and stability analysis of the optimization problems over $$\mathcal {K}_{1+}$$K1+.