Is there a better way to prove truthfulness of triangle's inequality for metric distance other than trial and error?

Question

I always get stuck at proving triangle's inequality when showing whether the function is metric distance .

Is there a better way to prove triangle's inequality. For example -

D(x,y)=|x-y|/(1+|x-y|) when x,y are in R

I am unable to prove triangle inequality for the above distance.

See https://math.stackexchange.com/questions/1145484/is-this-proof-sufficient-to-show-that-a-concave-function-of-a-metric-is-also-a-m — edm, Aug 21 '17 at 00:48

kimchi lover · Answer 1 · 2017-08-21T12:03:21.690

1

If $f$ is increasing and subadditive on $\mathbb R^+$, so $f(a+b)\le f(a)+f(b)$, then $f$ composed with a metric preserves the triangle inequality. Checking subadditivity of $f(x)=x/(1+x)$ is not very hard.

What I do first with such problems, however, is ask my computer to make 100 random points, and check all possible $100\cdot99\cdot98$ triangle inequality instances, numerically. This is a million checks, and if the proposed metric is in fact not a metric, I'll learn that fact quickly.

edited Aug 21 '17 at 12:03

answered Aug 21 '17 at 00:44

kimchi lover

24,981

Well is there a way to check it without using a computer ?? – Aashish Loknath Panigrahi Aug 21 '17 at 00:47
Checking that $x/(1+x)$ is subadditive is a few square inches of algebra, independent of any computer verification. Obviously, the computer check I described is a lazy man's way of sometimes avoiding having to think. – kimchi lover Aug 21 '17 at 00:52

Is there a better way to prove truthfulness of triangle's inequality for metric distance other than trial and error?

1 Answers1