[dgsb]

> Many classically trained pianists are excellent at improvisation.

I would not say that, the example you give are the most reknown composer of all times. I think they are more the exception than the reality of professional musicians. Not every one is able to reach such level of genius. It seems, proficiency in both style seems rather rare. I don't think I could name a single current world class contemporary jazz player which could reach the same level in classical music interpretation.

Regarding jazz improvistion, from my little experience of amateur player, jazz musicians mostly improvise together as a band, this is not a solo activity.