That depends on how you define what a "sports performance center" represents?
It may mean a training facility where specialized coaching and instruction takes place from experts in particular sports. In which case, knowing what sport you're interested in will affect the answer, as not all training centers support the same sports.
Or perhaps you mean a center where sports are performed. Again, this can be sport specific. Soccer and football are played on fields or in stadiums. Baseball and softball have special fields and a number of indoor sports such as Basketball and volleyball require gymnasiums. As such, the largest of any of these types of facilities in Murray, KY would be found on the campus of Murray State University - although they are not open to the public.
If you could be more specific with your question, a more specific answer can be given.
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