A convex real-function is a function such that all points above the function's graph form a convex set.
Why isn't it defined instead the opposite way: a function such that all points below it form a convex set?
Is there a particular reason for this?
edit: my question really is: Is there some kind of similarity that convex sets share with convex functions, but not with concave functions, which would explain why we use the same term for these different things?
I agree that this is odd. The words "convex" and "concave" have a meaning in ordinary English, and it's bad if their meaning in mathematics is something entirely different. In ordinary English, concave means hollowed out (like a cave, if you like), cupped, scooped out; convex means humped, rounded, hill-like, protruding.
The graph of a (mathematically) convex function is (English) convex if you view it from below, and (English) concave if you view it from above. One could argue that viewing from above is more natural. If you believe that, then the mathematical definition of convex is inconsistent with everyday English.
The same problem exists with convex sets. A (math) convex set is (English) convex when viewed from the outside, and (English) concave when viewed from the inside. Arguably, viewing things from the outside is the more common situation, so the choice in this case is easier to defend.