  We answer in the affirmative a recently-posed question that asked if there
exists an "untypical" convex mapping cone -- i.e., one that does not arise from
the transpose map and the cones of k-positive and k-superpositive maps. We
explicitly construct such a cone based on atomic positive maps. Our general
technique is to consider the smallest convex mapping cone generated by a single
map, and we derive several results on such mapping cones. We use this technique
to also present several other examples of untypical mapping cones, including a
family of cones generated by spin factors. We also provide a full
characterization of mapping cones generated by single elements in the qubit
case in terms of their typicality.
