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This is what I'm confused about. Cyprus is even closer, on clear days you can see turkey with the naked eye, and they didn't feel anything, atleast according to my grandpa. But Italy gets a tsunami warning?
Does alert mean there is a tsunami or just one is possible? >

Possible. Tsunamis are tricky, it's best to assume they're happening and then rule them out afterwards. >

Possibly a dumb question, but are tsunamis in the Mediterranean as big a threat as they are in the world's major oceans? 
In my head at least as the Mediterranean isn't nearly as deep and the volume of water isn't as great so the capacity for destruction might be lessened. >

Also: Why Italy? There’s this whole country (Greece) in between Turkey and Italy - wouldn’t that shield Italy from most or all tsunami threat?