What is the name of the rest of the triangle inequality

Question

The trianlge inequality states (for norms) that $$ ||a|| + ||b|| \ge ||a + b|| . $$ This can also be stated in terms of the quantity $$ r \triangleq ||a|| + ||b|| - ||a + b|| \ge 0 . $$ My question is the following: is there a standard name for $r$?

Answer 1

The norm is of course homogeneous. We have $$\frac{r}{2}=\frac{\|a\|+\|b\|}{2}-\left\|\frac{a+b}{2}\right\|.$$ For any real function $f$ (defined on a convex subset of a linear space) the quantity $$J(a,b;f)=\frac{f(a)+f(b)}{2}-f\left(\frac{a+b}{2}\right)$$ is called a Jensen gap of $f$ at the points $a,b$. Observe that $J(a,b;f)\ge 0$ for any $a,b$ implies that $f$ is Jensen-convex. Together with continuity (if some topology is given) it gives us even more: convexity in the usual sense.

So, applied to the remainder of a triangle inequality it could be called in this way. Original $r$ is of course the additive gap of a norm. Unfortunately, I have never seen such a name. The name Jensen gap is used by the researchers (in particular, one of my papers uses this name).