Geometry is indeed beautiful to learn.

It is beautiful yet not always easy… I am currently lost on Wikipedia where I try to understand the difference between a star domain and a radial set. There is certainly something I don't get in the definition of a "star-convex set". It's about 20 years I haven't manipulated these stuffs ! ^^;

I'd rather stay in one of the darker areas.

Interesting …
In the center, you can see anywhere yet anyone can see you.

Certain shapes, like that divot at the bottom, would cut out a large portion of the space as it would always be occluded from the left side. So if there was an optimum point it would have to be on the right side and not in the divot or along an edge which would quickly occlude the sides. Other than that, I can only imagine brute force is the only way to fully resolve the entire space. It's not a simple shape...

Sorry for the late answer. Today, I had the opportunity to discuss a little bit with my boss (who is also very good in geometry) and he proposed me another way of finding a decent "center" by considering the border as a polygon and accumulating the half-planar for each side or the (inside) area defined by each corner of the polygon. Not sure if my "explanation" is very clear.

Something a bit obvious (and I feel stupide now) is that the blue "pattern" is roughly a convex surface and, therefore, a gradient descent would be relevant… well, not exactly… as one may notice after giving a closer look at the blue "surface" that the "solution" is somewhere in this very noisy set (i.e. there are many local minima that would definitely make a primitive gradient descent completly unusable). Simulated annealing seems to be a powerful magic wand for many optimization problems.

Fun fact, maybe tomorrow or on Friday, a researcher working on CGAL (https://www.cgal.org/) may come to visit the lab where I work. I am not sure I would dare bothering him with this silly question (even if it is quite tempting).