Given a region $S$ in plane geometry, for any chord we can compute the average value of a function $f:\mathbb{R^2}\to \mathbb{R}$ over the chord. Function $f$ and its derivatives are continuous in region S (function f is a physical quantity, actually).
If we indicate with $\overline{f}_{C_1C_2}$ the mean value of $f$ over the chord $\overline{C_1C_2}$, we have $$\int_{\overline{C_1C_2}}f(x,y)=dist(C_1,C_2) \;\overline{f}_{C_1C_2}$$ The same for chord $\overline{C_3C_4}$, and so on.
Is it possible to find the value of $f$ for any point in the region $S$?
In the actual physical problem region $S$ is not always convex and may have holes, but if you think convexity makes the problem more affordable in a substantial way, you can assume it.
Follow up
I guess the inverse Radon Transform is what we need here.
If $S$ includes the boundary curve, then you can find $f$ on the boundary since you just take small chords converging to the boundary point. If $f$ is a physical quantity, then one would expect it to be basically infinitely differentiable... at least as a toy model of the problem.
Ignoring that, if $S$ is convex, then the question is answered. At stated in this paper: The X-Ray Transform. The X-ray transform of a function $f$ is gotten by collecting from $f$ only two pieces of data: for any point $x \in \mathbb{R}^n$ and any direction $\theta$, the integral of $f$ along the line starting at $x$ and having the direction $\theta$ (and going in the opposite direction as well to make an entire line). The paper cites a result that $f$ is completely determined by its X-ray transform if $f$ is differentiable and having compact support (i.e. $f \neq 0$ on a bounded collection of regions) and there is an inversion formula that gives you $f$ from the transform if $f$ is also infinitely differentiable.
The X-ray transform asks for the integral along a line and I do not see any straight-forward way to address this besides the region being convex. If you were doing this computationally, then maybe you could manually feed in the integral along a line as the sum of the integrals along the chords that make up where $S$ coincides with that line.
The X-ray transform is related to taking X-rays (duh duh dum), in that when taking X-rays we can only pass X-rays into a concealed region trying to find out the density of it.