From your post, it sounds like a self esteem issue to me.
I agree with spending as much time with your daughter as possible, talking, talking and did I say talking..... You can talk about everything and anything except the negative things about "spoiled little rich kid".
Get her involved in something she loves. When our daughter was 7 we felt like we had tried everything to find something she liked, ballet, soccer, etc. On a whim, she went to a martial arts class with a friend and that was it. She was hooked and she loved it. She is a black belt now and can hold her own with anyone. This was something we never would have thought about trying.
Try music? When our daughter went to 6th grade, it was a requirement to do 1 yr of music. The choices were orchestra, band or chorus. She picked Orchestra because she heard the director was great and had fun classes. Little did we know....she is very musically talented.. Who would have thought that??? Again, she is very talented and in the highest level orchestra at the school, planning to continue her violin through college.
Cheerleading. Nope, I did not think she would like this but she tried out and will be a high shcool cheerleader next year.
You never know what will interest them....just try to find something your daughter can take ownership of and accomplish something she is proud of.
Hang in there....there are a lot of "spoiled rich kids" in every area, every year....you have to learn to deal with everyone.
Good luck.