how to figure out plane convex domain if all length of its intersection with coordinate lines is known?

Question

Suppose $D\subset[0,1]\times[0,1]$ is a plane convex domain, define two function on $[0,1]$ related to $D$ as $$ h(x) = \text{length of line segment } D \cap \{ (x,y) | 0\leq y\leq 1 \} $$ $$ w(y) = \text{length of line segment } D \cap \{ (x,y) | 0\leq x\leq 1 \} $$

If the convex domain $D$ is given, for example $$ D = \{ (x,y) | a\leq x\leq b, f(x) \leq y \leq g(x) \} $$ then one can get the expression of $h(x)$ and $w(y)$.

My questions are

1. Is different convex domain $D$ result to different function pair $h(x)$ and $w(y)$?
2. If the answer of question 1 is true , how to figure out an expression of $D$ from the related function pair $h(x)$ and $w(y)$.

Answer 1

Consider the parallelogram $P$ with vertices $(0, 0), (1, 0), (2, 1), (1, 1)$.

Flip this about the line $y = \frac{1}{2}$ to get a parallelogram $P'$.

Then the $h$ and $w$ functions for $P$ and $P'$ are identical, but $P$ and $P'$ are different. (In particular, the $w$ function for each is $1$ for $0 \le y \le 1$, and $0$ otherwise).