Have not followed the myriad of links here yet. But I would like to add that core stability is one of the most important things, often overlooked in tennis, running and in general. Your core stability will affect the way (efficiency) you use your legs, your arms, everything.
Someone once provided a very illustrative image to me: "you cannot shoot a cannon from a canoe". That really drives home the importance of a stable core/rump for me.