Legal Commentary

For all these years, I've thought that a curve was concave if it looked like it wouldn't be able to hold water.  But it seems that, at least in some disciplines, whether something is concave or convex is a matter of perspective.  Something is concave if it curves away from you; something is convex if it bulges toward you. (These are the dictionary definitions as well.)

So what I had thought of as a concave curve is concave only if you imagine yourself standing below the curve looking up.  If you're above the curve looking down, it curves toward you, so it's convex.  When I said "concave," in other words, I really meant "concave from below," which is the same as "convex from above."  (Another possible perspective would be to imagine yourself standing at the origin, but this gets a little confusing, especially when the curve in question goes through the origin.)

As far as I can figure out, in the 1940s and 1950s, the more common approach seems to have been to talk about curves as concave or convex from above, sometimes even dropping the "from above."  As late as 1971, it was the view in at least one article that a curve like the one I drew above would be best described as "convex."  But I've only ever heard people talk about curves as concave or convex from below, usually dropping the "from below"--so the curve above would be described as "concave."  My mom, a math teacher, has the same view.  My mom and I both think, for example, that a twice-differentiable function that has a non-negative second derivative is by definition convex.

Perhaps the difference in terminology is no more than the difference between economics and math?  At any rate, I'm still a little confused and am curious to hear others' perspectives on this, especially people with background in different disciplines (econ, math, etc.), or with historical insights on the terms.