Hi Sol,
First of all thanks for the +1, and for the additional ideas you have on the matter. If I look at the original question, to me it clearly states that the density of the points is asked. The polygons are merely used to define which points should be combined to 1 center. On the other hand, we should ask Palermo what he wishes to achieve with the analysis, to be able to define what analysis suits his needs.
If I look close at the picture and the name of it ("Urban_density.jpg") it sounds to me that the polygons are counties and the points are urbanizations. To get the "center of gravity" it might be more realistic to use a certain characteristic as weight; say population per point. This would result in the center of gravity of population per county.
One more thought on using the outline of the polygon to determining the center; if you use for instance each vertex or equally separated points on the outline, a polygon with a lot of detail on one side (like the one center right in the picture) would pull the center of gravity towards the detailed part of the outline. I don't think this is what we want...
Kind regards,
Xander