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This large open space is the Palestra or workout area of the Stabian Baths. On the far side of the Palestra is the swimming pool, with auxiliary rooms on each side of it. On the left, south, side note the columns that supported a covered portico. On the right, north, side there were additional rooms near the women's area of the bath.
The Palestra was used for individual exercises as well as wrestling, racing, etc.
The Stabian Baths were the oldest and largest baths in Pompeii and were constructed in the second century B.C. They included a dressing room (apodyterium), a medium temperature room (tepidarium), a hot room (caldarium), and a cold room (frigidarium). One usually proceeded through the bath in that order. Besides these usual rooms, they included a men's and a women's sections (thus baths), a large open exercise area (palestra), and a swimming pool.
The earthquake of A.D. 62 severely damaged these baths and some areas were not in use at the time of the eruption of Vesuvius.