A **concave **surface curves inward. The word is easy to remember because a concave indentation in a wall makes a cave. A **convex **surface curves outward. Like many pairs of antonyms that are relatively rare and similar in sound, these two adjectives are easy to confuse.

*Concave *and *convex *are also geometrical terms; a concave polygon has at least one angle greater than 180 degrees, and a convex polygon is made of angles each less than or equal to 180 degrees.

## Examples

[O]ur resourceful Neanderthal cousins made sinks—or, rather, water basins—out of big rocks that eroded into a concave shape by centuries of rain. [Ottawa Citizen]

Long-sightedness can usually be corrected using convex (thinner at the edge than at the centre) lenses, which make objects appear larger. [Daily Mail]

What about those concave cheeks – are they natural or drug-induced? [Guardian]

Once trigger hairs are tripped by the prey, the plant bends its rubbery leaves into a convex shape, like a tennis ball. [CBC]

Some guys will chase anything that’s concave rather than convex in the appropriate location. [comment on Sydney Morning Herald]

Her contouring had the unfortunate effect of making her forehead look wildly convex, as if a toboggan were trying to emerge from it, fully formed. [Salon]

You should address whether a straight line is concave or convex or if neither, what is it–e.g., generalize concave to be non-convex.

“generalize concave to be non-convex”? Why did you waste time writing that? That statement was entirely redundant and blatantly obvious. Isn’t that what defines the very meaning of the two? Hence different nomenclature for different physical properties of opposite geometric structures. Last time I checked my math, straight lines in the most basic definition don’t have curves unless you consider different methods, i.e. parabola.

I raised the issue because spending time on wikipedia reading the pages for concave and convex (each term has multiple pages) and other sites, I find that there are different statements about whether a straight line is concave or convex, and arguments, and criticisms of various definitions. Your last sentence demonstrates my point–many people (presumably math literate) argue that a linear (or affine) function is concave (or convex), and talk about non-concave and non-convex functions in that context. So you need to go re-check your math.

I’m sorry my degree wasn’t in math. I guess I wonder, in space or on a plane, what causes a linear function to be concave or convex, or non-concave or non-convex. And excuse me, but I do not refer to Wikipedia as a reliable source, thank you very much. I just haven’t had Calc in 10 years.

The fact that people arguing about this on Wikipedia–but as much elsewhere on the web–emphasizes that even information on the web that sounds credible to a non-specialist always has to be verified as best one can with whatever sources are available. My BA is in math but that was back in the days before zero was discovered, and I moved on to a Ph.D. in computer science. There is no disagreement among mathematicians about the definitions of concave and convex, they are precisely defined on Wikipedia and elsewhere–and those can be verified by reading a mathematics textbook (even on the web). But the question of whether a linear (affine) function is concave is debated on Wikipedia, and mathematics textbooks address that. My recourse is to any contemporary mathematics textbook, which sides with the position that unless a linear function is a special case that is neither concave nor convex (which evidently introduces problems), there is a plausible argument to dichotomize functions as convex and non-convex, since evidently few people credibly argue that linear functions are convex. So that is the position I take, open to correction by real mathematicians, but none have done so, probably because they didn’t and won’t read my web site.

Wow, that’s incredible. What I don’t understand is why the academics don’t/won’t read your website. What you’re saying is truly amazing. I’m no math scholar by any means but have great respect for those who are, especially the stance you’ve taken in challenge. If I may, what is your website?