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 $r$?
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The triangular excess ? – Jan 25 '17 at 14:31
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It could be - not bad :-) – szw1710 Jan 25 '17 at 14:41
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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).
szw1710
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For a name "Jensen gap": this is my paper (I am 2nd named author): http://journals.bg.agh.edu.pl/OPUSCULA/31-1/31-1-07.pdf – szw1710 Jan 26 '17 at 17:59