I don't know what name if any is attached to the numbers I'm about to describe.

For a line segment, [a,b]

the number is 1 if for any k in (a,b)

and 2 if k=a or k=b.

For a square, [a,b] cross [c,d],

the number is 1 if k is in the interior

the number is 2 if k is on an edge

the number is 4 if k is a corner

For a cube, [a,b] cross [c,d] cross [e,f],

the number is 1 if k is in the interior

the number is 2 if k is on an face

the number is 4 if k is on an edge

the number is 8 if k is a corner

The concept I'm interested in might change these numbers if the spaces are non-rectangular. So,

For a trapzoid,

the number is 1 if k is in the interior

the number is 2 for the edges

at each corner number the number is inverse of the fraction of the angle of that corner compared to R^2.

Does this ring any bell for names that I can use for searching?