Cyrene.
Cyrene in Libya.
Antoich in Syria
No one expended the Greek empire because there was not a Greek empire. Alexander the Great, the king of Macedon, with allied Greek city-states conquered the Persian Empire. After his death his generals fought each other. This resulted in the division of Alexander's conquests into the Kingdom of Pergamon, the Seleucid Empire and the Ptolemaic Kingdom.
The capital of "New Rome" was Constantinople, formerly Byzantium, now Istanbul. The port city of Byzantium is at farthest southeastern point of Europe, adjacent to Asia Minor. The city was selected for its position controlling the Straits of Bosporus. Since the city was established under the instructions of the Roman Emperor Constantine I the Great, it was renamed in his honor as Constantinople and survived the Ottoman Empire (1922 AD) although since 1453 it had a second name given by its Ottoman Turk conquerors, which was Istanbul.
United Empire: Rome Western Empire: Rome then Ravenna Eastern Empire: Constantinople
Under Darius, the Persian Empire extended to the city of Taxila in Central Asia. He was emperor from 521 to 486 BC.
It is hard to know where they actually got control of. Possibly Bishkek in Kyrgystan.
The city of Jerusalem was within the Persian Empire.
Persepolis
Which city are you asking about. There were hundreds of cities within the Persian Empire.
baghadad
The Persian Empire versus coalitions of Greek city-states led first by Sparta, then by Athens.
The ancient Persian empire was located where Iran is today. Babylon was its capital city.
The Persian Empire and a coalition of Greek city-states.
The Persian Empire versus varying combinations of Greek city-states.
The Greek city-states occupied by, or threatened by, expansion of the Persian Empire in the eastern Mediterranean littoral.
Persepolis.