Which dance company shared the building with Joseph Pilates in New York City?

The New York City Ballet is the correct answer because it was one of the influential dance companies that shared a building with Joseph Pilates in New York City. Pilates famously developed his methods in the early 20th century, and his studio became a gathering place for dancers, particularly those from the New York City Ballet. His work significantly benefited dancers by enhancing their strength, flexibility, and overall performance.

The other dance companies listed do not have the same historical connection to Joseph Pilates. For example, American Ballet Theatre, Miami City Ballet, and Pacific Northwest Ballet were either established after Pilates had already created his methods or did not have significant interactions with him in terms of sharing a space. Understanding this historical context provides insight into how Pilates' techniques became integral to dance training in the United States, especially for the New York City Ballet dancers who sought to improve their physical capabilities.