Isn't gamma just the second derivative of the option px wrt to the px of the underlying? Long gamma position means second derivative positive, short gamma position means second derivative negative. The geometric interpretation of the sign of the second derivative is convexity.
All true. My question was about terminology - how to describe a smile-shaped function vs a frown-shaped function in mathematical terms. I corresponded last night with a mathematician who told me that "back in the day" when he learned calculus they referred to it as "concave up" and "concave down." He also confirmed what Nitro said - that today the preferred terminology is "convex functions" and "concave functions."
Yep, I remember my high school calculus. It's always been 'convex' or 'concave' (but that was in another, somewhat more math-heavy, country).