Why does “convex function” mean “concave *up*”?

Question

A function $f : \mathbb{R} \to \mathbb{R}$ is convex (or "concave up") provided that for all $x,y \in \mathbb{R}$ and $t \in [0,1]$, $$f(tx + (1-t)y) \le tf(x) + (1-t)f(y).$$ Equivalently, a line segment between two points on the graph lies above the graph, the region above the graph is convex, etc. I want to know why the word "convex" goes with the inequality in this direction, and how I can remember it. Every reason I have heard makes just as much sense applied to the opposite inequality ("concave down").

Answer 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!)

Answer 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

Answer 3

One of my professors told me the following memorable line: "A concave function looks like the roof of a cave." which helps me remember what is a concave and what is a convex function.

Answer 4

The primary concept is convexity, not concavity. It applies to geometric figures, originally lenses, and this usage was adapted to functions. There is no comparable concept of concavity for, say, 2-dimensional regions, except as the absence of the property of convexity. There is also no property for figures in general corresponding to the anti-convexity inequality, because most non-convex figures will be locally convex. It is a matter of historical convention that a function is called "convex" if the region above the graph of the function is convex, and it would have caused no mathematical problem to use the opposite convention based on the region below, but concavity is a more limited concept that is defined in terms of convexity (or only defined for functions) and not the other way around.

The terms "concave up" and "concave down" appear mainly in non-specialist US college textbooks on calculus. They are nonstandard terminology and, I think, bad practice that should be discouraged (with luck and sufficient ruthlessness maybe they can be squelched in a generation...). As far as I know the etymology went as follows:

1. Like "convex", the word "concave" has a prior use in optics. Concave (inward-curved) lenses are the opposite of convex lenses, so there is a pre-existing word for "not convex" or "convex in the opposite direction".

2. Convex has an absolutely entrenched mathematical use to denote convex figures as well as functions (and sequences) with increasing derivative.

3. Functions whose negative is convex occur frequently and "concave [function]" came into use as a convenient description of this situation. The linguistic logic was clear enough to make this immediately understandable. It's not clear whether it was more or less favored compared to statements involving the negative, such as saying that $-f(u)$ is convex, or $f$ is anti-convex, or that is it the negative of a convex function. I don't have data at hand from web searches or anything like that, but I think concavity is less common as a description of negatively convex sequences. For functions the ability to draw a graph makes the resemblance to lenses clearer so that both words seem sensible. (added: concavity as a counterpart to convexity for functions and sequences also gained momentum as its own term once log-convexity and log-convex became standard usage. Because the relationship between log-convex and log-concave functions is not simply change of sign but a multiplicative inverse, using only the words based on convexity might lead to confusion or circumlocution.)

4. Authors of US college calculus textbooks, writing for an audience not familiar with or necessarily interested in convex figures and optics, and aware of potential for confusion (e.g., the graph of a concave function still bounds a convex-shaped region, or the subsequent use of convex to describe functions of several variables and the regions on which those are defined) cooked up a terminology based on "concavity" as a stand-alone concept, limited to the one-variable context where $f(x)$ is graphed with the $y$-axis direction being upward. It's not clear how consistent this concave-up and concave-down terminology is between books and whether it agrees with the earlier, non-confusing use of concave to denote negative convexity.

Answer 5

A line is said to "support" the graph of a function (or indeed, any subset of the Cartesian plane) if it "holds up" the graph: that is, the graph lies entirely above or on the line. (After all, gravity pulls downward!) We might think of the union of all support lines as the "ground" on which the graph lies; everything else--its set-theoretic complement--is the "sky".

A function of the real numbers is convex if and only if its graph is the boundary between the ground and the sky. This is a special case of the more general idea of convexity that applies to arbitrary planar regions, the same as the familiar distinction between a convex and non-convex polygon, for example. For arbitrary regions there is no definite "up" and "down" anymore, though, so we say that a line supports a region when the region lies entirely within one of the two closed half-planes bounded by that line. (Thus, the interior and boundary of a convex polygon form its "sky" and everything outside is the "ground.")

In short, calling a "concave upward" function "convex" unites two closely related familiar concepts and is justified by the universal earthbound human experience that gravity usually pulls downward.

Answer 6

With the caveat that it's usually more helpful to devise your own mnemonics than follow someone else's

• here are a couple of mine, poorly drawn (the second is same as Srikant Vadali's answer):

• convex: smiley face

• Another way of remembering them, if you recall the meanings of convex and concave outside mathematics (as in lenses, etc.), is that you look from below: if the graph of the function viewed from below looks convex (i.e., bulging towards you) the function is convex, if it looks concave the function is concave.

• Yet another way is to keep in mind the definition: "a function is convex if its epigraph is a convex set". The epigraph is the set of points lying above the graph, and a convex set is one in which every line segment between two points in the set lies within the set. [Actually, for me, this definition is more useful for remembering what epigraph means :-)]

Answer 7

Instead of thinking about the graph of $f: \mathbb{R} \to \mathbb{R}$ as a 2-D object in the plane, think about $f$ mapping one number line onto another.

I'll return to that picture, just notice that there is plenty of "room" so to speak and that the arrows mapping point to image will never "fall back" on each other, overlap, or "crowd in" so long as the function is convex.

Imagine a closed loop in the plane whose interior is non-convex. It's like a deflated balloon. You need to "blow it up" until it's at least modestly ($\leq$) full of air for the interior to be convex. Similarly if you had a non-convex polygon and "blew air" inside of it, you would get a convex shape. So it's like convex shapes have to be sufficiently "inflated".

Similarly, the arrows in an $\mathbb{R} \to \mathbb{R}$-type picture like the above would be "flopping" inward onto each other, deflated if you will. In the convex mapping the arrows don't overlap at all -- they've got "pressure" or "energy" pushing them outward enough so that they don't overlap. So the image is properly inflated, if you will.

So @NateEldridge, the epigraph being a convex set is a red herring. Think about just the right-most point of a graph as it's being generated by a s-l-o-w graphing calculator. The image has to "outrun" the domain it comes from by $\geq$ each $dt$. And there you have your $f(\mathrm{interior\ of\ domain}) \leq \mathrm{image}_1 + \mathrm{image}_2$.

This is meant as an elaboration on @whuber's answer.

Answer 8

I don't know if this is the original reason for the terminology, but it makes a lot of sense when you look at optimization problems. Convex functions make minimization "easy", in the sense that if you have a convex function on a convex domain, it has a unique local minimum which is also the global minimum, and gradient descent works.

If you were typically looking at maximization problems instead, it would make sense to switch the meanings of convex and concave functions. But since many problems of practical and/or theoretical interest are naturally expressed as minimization (of potential energy, cost, area, etc.), we have "convex" defined the way it is.