A polygon compactness metric that filters out the noise of small concavities?

Question

I am trying to characterize the compactness of polygons. I have come across this definition of "compactness ratio" (perimeter/area), which works for most polygons, but I find as the negative concave space of a polygon decreases, the perimeter has a less meaningful role in the metric I am trying to characterize.

Specifically, I'm studying diffusion from various polygons. The literature shows that diffusion can be characterized by compactness. My problem is that the tighter concave spaces essentially have less impact on the diffusion because of interactions with the nearby boundary of the polygon. This isn't captured by the compactness ratio described above.

As an example, I've sketched diffusion contours near a tight concave space vs. a less-tight concave space:

Even though each of these two concave line strings have the same perimeter, the concave area on top has much less impact on the diffusion.

I'm thinking that the metric I'm looking at would be some ratio of the polygon's area to the area of a rounded buffer polygon. At some extent beyond the concavity, both of the sets of contours (or buffers) will approach a straight line, but it is evident that the top example will approach a straight line sooner (where the contours represent significantly higher values).

Is there such a metric that will characterize concavities which do not significantly influence diffusion? I can invent my own, but I'm wondering if another metric might already exist: Something that filters out the "noise" of the tighter concavities?

Update:

I'm realizing, that maybe all I need is the compactness factor of a simplified polygon (e.g., like this example).

Answer 1

First of all, to get a dimensionless value, you need to use perimeter^2/area.

Second, in addition to concavities, your diffusion is probably affected by sharp angles.

This suggests that what you are actually after is the differences between the actual perimeter and that of the "slightly smaller" and "slightly larger" polygons (defined as the normal expansion/contraction of the polygon by some characteristic length).

Answer 2

In case if an answer to this old question can be still useful for someone, there are other shape metrics which use not only area and perimeter. Namely, which use only area because it is the perimeter which causes problems.

There are Related Circumscribing Circle index which is a ratio of the area of the polygon to the area of its minimal enclosing circle (area/circumcircle area). And more useful in this case may be Solidity or Related Convex Hull index which is a ratio of the area of the polygon to the area of its convex hull (area/convex area).

There also may be something useful in this question on the parallel SE site. And yes, generalization of polygons will be very useful.