Name the book in which Peter Brook makes a study of late 1940s theatre.

The correct option is The Empty Space.

Peter Brook, one of the most influential theatre directors of the 20th century, published his seminal book The Empty Space in 1968. In this work, Brook categorizes and analyzes theatre into four distinct types: the Deadly Theatre, the Holy Theatre, the Rough Theatre, and the Immediate Theatre.

The book draws extensively from his practical experiences and observations of theatrical productions, trends, and developments, including a study of late 1940s theatre and subsequent decades, to explore what makes theatre vital and meaningful.