I am playing a violin concerto with an orchestra and a fairly large audience in a few weeks. So my question is, in the long Tutti parts where I don't play, should I face towards the audience, and what should my posture be like?
Some performers prefer the "look busy" approach. I once attended an oboe concerto performance in which the oboist swabbed out his instrument every time he had more than 4 measures' rest. I suppose a violinist could rub down excess rosin, pretend to tweak the bow tension, re-set the comfy rag under the shoulder rest, etc.
In any case, looking attentive, and keeping an eye on the conductor, are good suggestions. Then about 2 bars before your next entrance, turn to the audience, close your eyes as though in rapture, and dramatically prepare your posture and bow for the entrance.
You could remain directly facing the audience, with the smile gradually dripping from your face to be replaced by something between boredom & abject terror...
...or you may find it a lot easier to just quarter-turn & watch the conductor.
You're still up there, looking professional, paying attention, but aren't being subjected to the actinic glare of the audience's gaze whilst having nothing to actually do.
You could look at your feet... but it doesn't come over well ;)
It would be absolutely rude and unprofessional not to focus on the music when anybody but yourself is playing. You are a soloist, not the sole player. The others aren't just fluff between your parts.
And even if you think the audience has only eyes for you, then you should use this to make them aware of the music rather than engaging in a private conversation at the cost of other musicians.
See a primadonna handle this situation professionally: the first minutes of this video of Maria Callas singing Carmen are purely instrumental. She stays focused on the music, staying in character, not really engaging with the audience and when her actual part comes, she not jumps into activity and then stops again but that part comes as a natural continuation of her posture during the long tutti parts.
Of course, operatic singing cannot be compared with instrumental soloing since the former is supposed to convey a character, but still the situations are not entirely dissimilar.
My go-to approach for this is just to let your eyes guide the audience to where you believe their attention should be. If you look at the orchestra after you have been the center of attention for a bit, they'll do the same thing. Lead the audience to where you want them to be...both with your music, and with your body (and eye!) language.
Edit: Once it's your "turn" again, look out into the audience to command their attention.