I have been thinking about this thread quite a bit. I will admit that I am not in the same kind of shape as I would have as my ideal, but physical fitness is very important to me none-the-less.
I do feel, as TimR mentioned, that choices need to be made along these lines as I don't see myself being able to spend 3 or 4 hours working out each day (somethng that sounds enjoyable to me) as well as fitting in the kind of quality practicing, teaching and research I want to be doing (not to mention normal life stuff).
It is mainly a matter of looking hard at one's priorities. Piano is definitely my main priority. Because of this, I often aim to have the other things that I do in life serve this priority in some way. Afterall, I am bringing my entire person to the art, so it can only help to refine as an individual. Two main examples come first to my mind, although there are many more that I am sure you are already thinking of.
1. Interaction with others. Because communication is very important to me, and becuase music is communicative, I am very conscious of how I communicate with people in all venues. I try to be aware of how I am saying things and how this may make the other person(s) feel. I of course do not always say the "right" things, but, I also learn from this.
My point here is that this kind of attention if done with awareness, will indeed greatly serve my main priority; piano. It is necessary as a matter of fact (for me). I am writing this very post to help clarify these ideas in my head which will in turn help me as a musician, as well as help me to pass ideas on toward my students.
I feel the same is true with everything we may do, athletics are a fabulous example. Having been thinking on this thread, I went swimming yesterday to get in a workout. I needed to clear my head after having spent the whole day with study, I also just needed to experience some good and thorough physical exertion.
Here is what was most interesting to me about this:
As I began, I thought I would not make it even 20 minutes as I had been exhausted before even entering the pool. After a few laborious laps however, I decided to think on the subject of one of the fugues I had been working on earlier. This changed everything! I began to swim within the rhythm of this fugue. I felt I was moving with the music. My workout turned into pure joy, and lasted far longer than any workouts had ever lasted previously to it. I never ran out of energy and stopped only in decision to get back to my study. The workout had already fulfilled its purpose for the day.
This is what I walked away with:
1. I had gone through and memorized in my mind the gist of the formal structure of this particular fugue.
2. I have a very clear subject in my mind now.
3. My body learned to move with the music in ways that I cannot explain easily in words (and I don't feel like trying) but of which are entirely practical and applicable to my study at the piano.
4. I got one of the best workouts of my life during which I could maintain a more steady tempo, developed better form, and seemed to awaken energy more than expend it.
5. My mind was more clear and fresh in general, leaving me ready to resume my studies.
6. etc. etc. etc.

The bottom line here is (no pun intended), that I think there are vast and direct benefits of physical fitness toward playing the piano, as well as the acts invovled with getting it and maintaining it.
m1469 Fox