Sphere of influence
A Sphere of influence is generally claimed by large or imperial powers over other countries or territories within a certain radius. It usually means that the superior power insists upon recognition by those countries within this 'sphere' as someone who must be consulted on all important issues.
Possibly the best example is the Munroe Doctrine a United States of America foreign policy position that opposes European colonialism in the Western Hemisphere and holds that any intervention in the political affairs of the Americas by foreign powers is a potentially hostile act against the United States. According to the Doctrine the Old World (Europe) and the New World were fundamentally different, and should be two different spheres of influence. The United States, for its part, would not interfere in the political affairs of Europe, or with existing European colonies in the Western Hemisphere. (This was ignored when the USA declared war against Spain and invaded her colonies of Cuba and the Philippine islands.) The doctrine was central to USA grand strategy in the 20th century, when, however, they again ignored the Doctrine to enter into European wars and politics and the occupation of parts of Europe.
Russia has an unwritten sphere of influence over all those neighboring state which were once pat of Tsarist Imperial Russia.
During the 20th century the Empire of Japan also sought to extend its sphere of influence against Korea, Imperial China, Imperial Russia and the German Empire. Their defeat in World War II brought this to an end.