# why are convex functions defined as functions who's epigraph is a convex set?

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?

• Well, we'd have to rewrite all of the other definitions of convexity if we did that, too. For instance, twice differentiable functions would need negative semidefinite Hessians, and the secant test would need the direction of its inequality reversed. Apr 12, 2017 at 12:15
• We would also lose some nice conceptual analogies. For example, if I consider a physical model of a 2D convex function, and I were to drop a ball somewhere on its surface, then it will tend to move towards the minimum due to gravitational pull, no matter where it is placed. Not so with what we call a "concave" function, and what you are proposing hypothetically to call "convex". Apr 12, 2017 at 12:19
• And our examples of "canonical" convex functions would change to more verbose and inelegant alternatives: $f(x)=-x^2$, $f(x)=-|x|$, and so forth. All we'd have is the lowly logarithm. Eventually people would throw up their hands and say "can we just ditch all of the negative signs here and switch our definition around?" Apr 12, 2017 at 12:20
• I don't see the conceptual analogy with the ball example. Why not simply call a function where such a ball would go to the minimum, a "concave" function? If we would switch concave and convex everywhere, how would that make the conceptual analogy dissapear? i.e. why not instead of writing negative signs, simply speak of those canonical functions as concave functions? Am I missing something in your point? Apr 12, 2017 at 12:26
• Why not call convex sets concave sets? Apr 12, 2017 at 12:27