An effective weighted K-stability condition for polytopes and semisimple principal toric fibratons

The second author has shown that existence of extremal K"ahler metrics on semisimple principal toric fibrations is equivalent to a notion of weighted uniform K-stability, read off from the moment polytope. The purpose of this article is to prove various sufficient conditions of weighted uniform K-stability which can be checked effectively and explore the low dimensional new examples of extremal K"ahler metrics it provides.