Here is example of some planar lamina; non-convex, but without self-intersections. I'd like to define some "my" center of lamina that will be always and only inside it. Is it possible to do it correctly?
For instance there are two well-known centers:
1. Point that minimizes the sum of squared Euclidean distances between itself
and each point in the set - mass center.
2. Point that minimizes the sum of Euclidean distances - median.
With Mathematica we can find points that minimize the sum of some other distances - Manhattan distance (point 3) etc. But I just wish to find a centroid close to red point. Is it possible to do it correctly and analytically?
The idea of dtldarek about inside paths was a really good proposal. And the only way I see for to find such paths is covering of lamina with some network graph. There are many methods and software for exploring it.
"Mathematica" has a lot of graph functions, so I spent some time conducting experiments. We may choose any suitable network, but we must make a weights for edges with Bray–Curtis dissimilarity. Than we can find a center that minimized sum of graph distances to other points.
Only this method gives results looking realistically to me for any areas:
Why Bray–Curtis? I don't know.