Why is the convex function called the concave function in the mathematical concept?

Some institutions in the Chinese continental mathematical field the definition of function convexity differs from others.

So let's talk about convex functions (convex function) for what is called convex (convex):this is because the convex function is associated with a convex set (convex set), while the definition of a convex set is not controversial.

1. The convex function and the convex set are connected by the concept of sublevel sets.

First, look at the sublevel sets of a function. For a function, its-sublevel set is defined like this:

That is, in the function definition field, the corresponding function value is less than the value of the argument of the set of values.

Contact 1:For any case, the-sublevel set of a convex function is a convex set.

Note that the inverse proposition of the proposition is not tenable. To get a better understanding of the concept and why the inverse proposition is not tenable, let's look at an example (figure from resources):
This is a function that defines both the field and the domain, which is non-convex. Its-sublevel set is, obviously, a convex set. We can even see that for this function, given any of its-sublevel set is a convex set, but this function is not a convex function. Such a function has a name called Quasiconvex.

2.the convex function and the convex set are connected by the concept of epigraph.

Then look at the epigraph of a function. It is defined in this way:

This prefix epi seems to be the meaning of above, so the meaning of epigraph is probably "above figure". For a function, its epigraph should be a subset.

Then look at the chestnut just now, the epigraph of this function is the gray part above the function (forgive me):
See, this is not a convex set.

Contact 2:The epigraph of a convex function is a convex set, and vice versa. In other words, a function is a convex function,when and only ifIts epigraph is a convex set.

Resources:
Stephen Boyd and Lieven Vandenberghe. Convex optimization. Cambridge University Press, 2004.

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Suddenly found Sublevel sets and epigraph the two concepts are also relative, the sublevel sets definition of the less than equals to the number of greater than equals may have "Superlevel sets", the same can also define "hypograph", so Perhaps the difference between bump or by convention bar?

from:http://www.zhihu.com/question/20014186

Why is the convex function called the concave function in the mathematical concept?