On this promontory was located the private wing of Herod's palace. The palace extended eastwards (you are looking west in the photo below) to the administrative wing of the palace with the north-east corner juxtaposing onto the hippodrome-amphitheatre.
The private wing of the palace features a 35m by 18m by 2m deep rock-cut pool, with a square base (center in the photo below) in the centre where it probably supported a statue.
This building became the residence of the prefects and procurators who governed Judea after the death of Archelaus (4 BC-6 AD). It was very likely that it was in the courtyard of the administrative wing of this palace that Paul was brought before Felix (Acts 23:35; 24:24ff) and Festus (25:6-12; 23ff) for his case to be heard before he appealed to Caesar and was sent on to Rome.