This process was aided by the universal adoption of Greek as the language of intellectual culture and commerce in the Eastern Mediterranean Sea, and of Latin as the tongue for public management and forensic advocacy, especially in the Western Mediterranean.
Though the Greek and the Latin never became the native idioms of the rural peasants who composed the great majority of the empire's population, they were the languages of the urbanites and cosmopolitan elites, and the lingua franca, even if only as corrupt or multifarious dialects to those who lived within the large territories and populations outside the Macedonian settlements and the Roman colonies. All Roman citizens of note and accomplishment regardless of their ethnic extractions, spoke and wrote in Greek and/or Latin, such as the Roman jurist and Imperial chancellor Ulpian who was of Phoenician origin, the mathematician and geographer Claudius Ptolemy who was of Greco-Egyptian origin and the famous post-Constantinian thinkers John Chrysostom and Augustine who were of Syrian and Berber origins, respectively, and the historian Josephus Flavius who was of Jewish origin and spoke and wrote in Greek.
Based on the above definition, it can be confidently asserted that the "cores" of the Greco-Roman world were Greece, Cyprus, Italy, the Iberian Peninsula, Asia Minor (roughly corresponds to modern Turkey), Gaul (modern France), Syria, Egypt and Africa Proper (Tunisia and Libya). Occupying the periphery of this world were "Roman Germany" (the Alpine countries and the so-called Agri Decumates, the territory between the Main, the Rhine and the Danube), Illyria and Pannonia (the former Yugoslavia, Albania, and Hungary), and Moesia (roughly corresponds to modern Bulgaria).
Also included was Dacia (roughly corresponds to modern Romania), Nubia (roughly corresponds to modern northern Sudan), Mauretania (modern Morocco and western Algeria), Arabia Petraea (the Hejaz and Jordan, with modern Egypt's Sinai Peninsula), the Tauric Chersonesus (modern Crimea in Ukraine).
The above seems to ignore the major rivalry between the Greco-Romans, during their period of ascendancy, and the great empire to the east, the Persians. See Xenophon,The Anabasis, or, the March Up Country, the Greco-Persian wars, the famous battles of Marathon and Salamis, the Greek tragedy "The Persians" by Aeschylus, Alexander the Great's defeat of the Persian emperor Darius and conquest of the Persian empire, or, the later Roman generals' difficulties with the Persian armies, such as Pompey the Great, and of Marcus Licinius Crassus (conqueror of the slave general Spartacus), who was defeated in the field by a Persian force, and was beheaded by them. (Ref: Appian, The Civil Wars).