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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?

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  • $\begingroup$ 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. $\endgroup$
    – Michael Grant
    Apr 12, 2017 at 12:15
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    $\begingroup$ 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". $\endgroup$
    – Michael Grant
    Apr 12, 2017 at 12:19
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    $\begingroup$ 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?" $\endgroup$
    – Michael Grant
    Apr 12, 2017 at 12:20
  • $\begingroup$ 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? $\endgroup$
    – user56834
    Apr 12, 2017 at 12:26
  • $\begingroup$ Why not call convex sets concave sets? $\endgroup$
    – Michael Grant
    Apr 12, 2017 at 12:27

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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.

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  • $\begingroup$ Perhaps you can offer more insight into what these words mean in ordinary English. I'm not seeing how you've applied the English definition to functions. To sets it seems more straightforward. $\endgroup$
    – Michael Grant
    Apr 12, 2017 at 12:57
  • $\begingroup$ That said I don't agree that a convex set looks concave when viewed from the inside. It looks like... I'm inside a convex set ;-) $\endgroup$
    – Michael Grant
    Apr 12, 2017 at 12:58
  • $\begingroup$ Yes, you're right. I was thinking of quasiconvex, and I even botched that. I'll remove that last paragraph. $\endgroup$
    – bubba
    Apr 12, 2017 at 14:08

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