I apologize beforehand for the vague title and the length of the description I am using to setup my question; I can't seem to be more concise without sacrificing clarity.
Call a region in the plane "nice" if its intersection with any line consists of a finite number of closed intervals. (Is there an existing term for such a region?). The "size" of any such intersection is, naturally, the sum of the lengths of these intervals.
Consider a nice region R. The "cross section of R along the z-axis" is the (nonnegative) function h given by:
C(t) := size of the cross section of R at the (horizontal) line z = t
Similarly, we can define the cross section of R at an arbitrary line.
Here are a couple of questions with (almost) obvious answers:
Question 1: Given a nonnegative function h, does there exist a nice region whose cross section (along some line) is h?
Answer 1: Yes, always. (easy)
Question 2: Given two nonnegative functions g, h and two lines k, l, does there exist a nice region R whose cross sections along k, l are g, h (respectively)?
Answer 2: Depends. Not hard to come up with examples for both possible answers.
Question 3: Given two nonnegative functions g, h and two lines k, l, what are sufficient (but not overly restrictive) conditions that guarantee the existence of a nice region R with the prescribed cross sections?
Answer 3: I don't know ... please help!
Now consider a similar situation in 3 dimensions. Define a region R to be "nice" if the intersection of any plane with R is (measurable and) of finite area. We can define the cross section of a region R along an arbitrary line analogously to the 2d case.
Question 4: What are sufficient conditions on g,h,k,l that guarantee the existence of a nice region (in 3d) with the prescribed cross sections? What if there are more than two prescribed lines and functions?
Answer 4: I have no idea! This is really the question I'm interested in. Any ideas, suggestions or references would be appreciated.