This field is not my meat, but let me try to answer according to my understanding.
This is the intersection of the $(+,+,\dots,+)$-cone of your domain space with a (translated) linear subspace thereof. For instance, if the domain space is just the plane, you’re talking about either a single point in the first quadrant, or the first-quadrant part of a line, or all the first quadrant.
If the domain space is $\Bbb R^3$, you’re looking either at a single point in the first octant, or the first-octant part of a line, or the first-octant part of a plane, or the all the first octant.
I don’t know what the book means by “limited”: if they mean compact, that is certainly not the case; if they mean closed, that is all right.
The set defined here turns out to be convex, but you are making a mistake if you are trying to see this definition as a modification of the book’s definition of a convex set. I would say that there are no hyperplanes specifically mentioned here, though if you search for them, you will find them.