I'm reading this paper http://www.math.hawaii.edu/~erik/papers/cat0-A.pdf and it looks like I don't get one point. It's the remarks under definition 2.2., mostly the sentence:
,,Imagine a vertex $ x $ in the ambient $ \mathbb{R}^N $, all of whose coordinates extend $ n $. The interstection of the interval from $ x $ to the origin $ 0 $ with the ball of radius $ n $ is an $ N $-dimensional tetrahedron containing $ {n + N \choose N} $ points of $ \mathbb{Z}^{N} $''
By intervals we understand integer points of polyhedrons. For instance, if $ N = 3, n = 1 $, then the interval from $ x = (2,2,2)$ to the origin is a hexahedron $ [0,2]^3 \cap \mathbb{Z}^3 $.
I don't get why the intersection of such interval with a ball has to be a tetrahedron with $ {n + N \choose N} $ integer points. I would appreciate some explanation