In all likelihood your "fat face" and puffy cheeks are genetically determined – particularly since you say that this look runs in your family. I doubt that your liver function has anything to do with the shape of your face unless your cheeks swelled suddenly, which does not appear to be the case.
As long as you are not overweight, there are no changes you can make in your diet or exercise habits to affect the shape of your face. Nor do I know of any dietary supplements that would make a difference. I've seen Internet promotions for facial exercises said to reshape the face, but there is no medical evidence that they work.
I'm not a big fan of cosmetic surgery, but it does offer the only dependable way to reshape the face. A procedure called buccal fat pad excision (or, more simply, cheek reduction surgery) removes the facial fat that is responsible for puffy cheeks. The incision is made inside the cheeks. If this is something that interests you, bear in mind that with age everyone loses facial fat. Take a look at your relatives who had puffy cheeks when they were young. If their faces have thinned, you may be headed in the same direction, and surgery to remove fat from your face now could lead to a gaunt look later in life. If you want to consider surgery, be sure to consult a plastic surgeon who is very experienced with the procedure.
However, before you think about something as drastic as surgery, you might ask a good hair stylist about the most flattering cut for your face. A fashion stylist might have ideas about the kinds of necklines that would minimize your facial fullness, and a make-up artist might give you some contouring tips that would help as well.
Andrew Weil, M.D.