Convexity, concavity and...Nonconvexities

Hi all,

I get confused about these three things. My understanding is the
following:
1. convexity and concavity refer to the shape of the function. So
f''(x)<0 for a concave function (<= if quasi-concave), and f''(x)>0 for
a convex one.
2. "nonconvexities" refer to set theory, and the question there is
whether the linear combination of 2 points in the set is also in the
set.

But now I'm reading an article by Romer talking about "nonconvexities",
in which he seems to define nonconvexity as F(aX,aY)>af(X,Y). But this
sounds to me awfully like a definition of "convexity" (in the sense
that it is not concave).
Then that means that nonconvexity=convexity.... This is confusing!

Thanks for your help!
Tom

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