Why does "convex function" mean "concave *up*"?

Solution 1:

Not sure why convex is defined that way, but one way to remember is that the derivative is monotonically increasing for some convex functions.

Or maybe just remember that $e^x$ is conv$e^x$. (I just thought of this one!)

Solution 2:

Lets say that you accept the definition of a convex set in higher dimensions, like a sphere in $\mathbb{R}^3$. The question I seek to provide insight into is why convex functions in one variable are defined as opening up instead of down, since this seems like an arbitrary definition. This is because, depending on how you look at the graph, you could naively view the function as bending outwards (like a convex set) or inwards (concave). However, there is a nice connection between these two things using metric spaces that I think can provide some meaning to the way it is defined.

Most of the metrics that you are familiar with have open balls that are convex, such as the standard metric. But some are actually non convex. A good example of this is $ d(x,y) = \sum \sqrt{|x_i - y_i |} $. (note that $\sqrt{x}$ is not a convex function)

Here is an interesting condition:

Given a metric $d$. If for all $y,z\in E$ and $0\leq t\leq 1$,

$d \left(x, \ t y \; \, + \; (1-t) z \right) \quad \leq \quad t d(x,y) \; + \; (1-t) d(x,z) $

then the open balls formed by $d$ are convex. [1] In other words, if you fix $x$ and $d(y):\mathbb{R}^n\rightarrow \mathbb{R}$ is a convex function, then the open balls are convex sets.

Usually $d(x,y) = \sum f \, (x_i,y_i)$, for some $f:\mathbb{R}^2\rightarrow\mathbb{R}$. If we fix $x$ and $f:\mathbb{R}\rightarrow \mathbb{R}$ fits the definition of a convex function, then $d$ will also be convex, and the condition will be satisfied, giving us convex balls.

So convex functions (if they can form a metric) will give you convex open balls. A nice connection that makes the definition make more sense. Other conditions that guarantee convex open balls are discussed in the paper I reference.

[1] Norfolk, T. (1991). When does a metric generate convex balls? www.math.uakron.edu/~norfolk/convex.ps