Yoga's principle of ahimsa, or non-violence (perhaps better translated as "un-violence," a refraining from all forms of harm) is the watchword here. When we practice ahimsa toward ourselves, we respect the limitations of the body and do not push ourselves to the point of injury. We can also see how feelings of competition with others lead to our own unhappiness. When we practice ahimsa toward others, we do not wish to see them harmed, which leads to the yogic principle of mudita, or taking delight in the virtue of others. It's pretty much the opposite of schadenfreude. Maintaining the equanimity of mind that yoga brings can help the dancer safely navigate a world of competition.
The following video is a very basic series of poses for the dancer. I've heard some ballet teachers specifically recommend NOT doing yoga because there is so much work in parallel that there is a fear dancers will lose their turnout. I've included two poses done with turnout here (vrkasana and eka pada rajkapotasana) to demonstrate that yoga can indeed be used to improve turnout, but there are also plenty of parallel poses as well. In order to prevent injuries and maintain healthy joints, the hips should be worked from ALL positions-turned out, turned in, and parallel. Doing yoga can bring balance to dancers' hips that may be tight from being constantly turned out. I've included poses here designed to strengthen the core, build balance, and stretch the hips; all things that are imperative no matter what style of dance you do. All my videos are ten minutes long so I can put them on YouTube, but please feel free to hold the poses for longer than I do here. I also recommend doing a full warmup and a few sun salutations to build heat in the body before performing the sequence below. Namaste!