I too would like to extract information from a polygon and am having trouble. Assume the polygon outlines a county or parish. For example;

GeoGraphics[{Edgeform[Black],GeoStyling["ReliefMap"],Polygon[East Baton Rouge (administrative division)]},GeoBackground->None]

generates a pretty cool outlined releif map. Double clicking inside the map highlight what must be the nodes of the polygon. The heights represented inside the relief map must have location data (Lat, lon?) to know where to illustrate the point.

To continue with my efforts I would like to use those locations, perform some calculation with the representative height and present the result instead of the height.

I cannot figure out how to extract the lat, lon or X,Y data. Any help would be appreciated. JAC