Primates are not very mobile. Here looking at typical monkey ranges

Allan Krill

Chimps are our closest relatives, and gorillas next closest. They live along the Atlantic Ocean, far from the Danakil highlands, the Red Sea, and the Indian Ocean. They have been living near the Atlantic Ocean and evolving in the places they are now for millions of years. 

Probably every species of primate (except human) has been where it now is for something like the past million years. Look at the geographic range of monkey species.  Here is a list of over 100 species of Old World Monkeys. 

You can click on any of the species names on that list, and see a map that shows the geographic range of that species. Here are the first 10:'s_Swamp_Monkey_area.png

And here are the geographic ranges of the great apes: chimpbonobogorilla, and orang.

These range sizes must be similar to the range of the primate that was the ancestor of humans (before acquiring the big brain that let our ancestor expand its range to many environments.) That primate probably inhabited a limited range with a single similar habitat, like these others. That range was probably near the chimps. 

Genetics shows that humans are closely related to chimps, and next closest to gorillas. It is not just genetics: the 4 taxa of chimp also look different.  See the table here: Appearance  

Conclusion: The Atlantic Ocean (not the Indian Ocean, Red Sea, or Mediterranean Sea or Tethys) is where the human ancestor got its big brain and then became cosmopolitan. 

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