According to Eric Dubay, some early modern sailors tried to circumnavigate Antarctica, and it took them a lot longer than it should have, so for this reason, among others, he figures Antarctica is the periphery of our plane, not the center.
He also said the south pole is a lot colder, darker and lifeless than the north pole, which would be the case if the south pole was the periphery, because the sun would be further away from it than the center.
He and other flat earthers may have other reasons, but those're the ones I'm familiar with.