Fitness center
'fitness center' is the definition.
(I've seen this in another clue)
This is all the clue.
(Other definitions for gym that I've seen before include "Area for physical exercise in short" , "Place to get fit" , "Physical education" , "Briefly, a keep-fit place" , "Fitness centre" .)
(Other definitions for core that I've seen before include "Heart of the matter" , "Crux" , "Central or most important part of something" , "Kernel" , "The innermost part" .)