# Numbers associated with boundaries of manifolds

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?

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The think my question is related to the Lebesgue's density theorem. – user6137 Mar 3 '11 at 19:28
You're measuring solid angles. Technically you're looking at the ratio of the solid angle (content) of the "link" of the stratum and the corresponding solid angle of a point in Euclidean space of the same dimension. – Ryan Budney Mar 3 '11 at 19:44
I'm only measuring solid angles at the smallest non-empty boundary, not in general. – user6137 Mar 3 '11 at 20:03
Well, you haven't said what it is you're doing. The solid-angle interpretation is consisting with all your examples so perhaps provide an example that disagrees with this interpretation. – Ryan Budney Mar 3 '11 at 20:15
"solid angle" can be interpreted in any dimension, in any Riemann manifold with corners. – Ryan Budney Mar 3 '11 at 21:46