How do I perform a find using $geoIntersects / 2dsphere in an array subfield?

I have a collection with documents and each has a field array with polygons.

I want to test if my polygon geoIntersects any of those polygons. They are all box-shaped if it helps.

I will add bounty points if you add in the possibility of having my polygon in that array I'm testing. I wouldn't want it intersect with itself...

Cities:

{_id, buildings:[ {coo:{shape:"Polygon", coordinates:[ [0,0], [4,0], [4,5], [5,0],[0,0] ] }, {coo:{shape:"Polygon", coordinates:[ [0,0], [4,0], [4,5], [5,0],[0,0] ] } ] }}

Imagine I have around 100 or so elements in the array. I want to determine if, for a given shape, that might be included inside the array, if it intersects any or none of the other polygons (again, excluding itself it is in there).