Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|improve this question
    
The think my question is related to the Lebesgue's density theorem. –  user6137 Mar 3 '11 at 19:28
2  
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
3  
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
4  
"solid angle" can be interpreted in any dimension, in any Riemann manifold with corners. –  Ryan Budney Mar 3 '11 at 21:46
show 1 more comment

1 Answer

up vote 0 down vote accepted

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.)

share|improve this answer
2  
Right . . . hence Ryan's first comment above. –  aaron Mar 3 '11 at 21:41
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.