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?



Numbers associated with boundaries of manifolds
I was calculating the approximate density of a set A (line segment, square, cube, etc) in a ε-neighbourhood of a point x. It's related to the Lebesgue's density theorem. (Actually I was calculating the inverse of these numbers.)


