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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.

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  • $\begingroup$ Most likely you can define "nice" as compact as each closed interval has an open cover, and thus it would have a finite amount of open covers. $\endgroup$ Mar 12, 2017 at 19:01
  • $\begingroup$ Also I'm not sure Q1) is yes, always. But I am also unsure what you mean by it too. Are you asking that we can find a compact cross section for all non-negative functions, or that given a function I can find a section of it that is compact? The former I can find plenty of examples of no, and some fun ones of "Oh hell no!" For the later yes that is true. $\endgroup$ Mar 12, 2017 at 19:24
  • $\begingroup$ Despite my best efforts to be clear, I might have misled you. I'm trying to avoid consideration of the monstrous sets that typically get mentioned in measure theory. Here is what I had in mind. Given any nonnegative function h, define a function g by $g(t) = \sqrt{h(t)/\pi}$. Then let $R_0$ be the solid bounded by the surface generated by the graph of g as it is revolved around the x-axis. Let $R$ be the result of rotating $R_0$ by 90 degrees in the y-axis. I believe that this answers question 1 in the 3 dimensional case. The 2 dimensional case is even simpler. $\endgroup$
    – sitiposit
    Mar 12, 2017 at 19:54
  • $\begingroup$ Sorry for taking so long for a response. What happens if we let $h: \mathbb R \to \{\vec x \in \mathbb R^n:x_i \geq 0,\: \forall i\leq n\}$ such that $h$ is bijective? Would this have a "nice" region R? $\endgroup$ Mar 12, 2017 at 23:49
  • $\begingroup$ By a nonnegative function h, I meant $h : \mathbb{R} \to \{x \in \mathbb{R} \ | \ x \ge 0\}$. $\endgroup$
    – sitiposit
    Mar 12, 2017 at 23:52

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