The same for the public transport -- in cities where rich
    people use it, the service is better

I wonder which is the cause and which is the effect here. Maybe some cities were proactive in improving their safety and public infrastructure, and as a consequence have attracted rich people to live there. That's really the only way it will work, anyway - getting anything done in the civic sphere takes a lot of time and effort, and it's not worth it for most people and companies. If cities wish to attract them, they need to proactively improve themselves.