Given the reverse triangle inequality:
$$ |x+y| \geq ||x|-|y|| $$
Is it valid to extend this to three terms like so:
$$ |x+y+z| \geq ||x|-|y|-|z|| $$
If this is correct could someone point me in the right direction for a proof? Thanks heaps.
2026-03-31 07:48:03.1774943283
Reverse Triangle Inequality with Multiple Terms
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It is false, take $x=y=1$ and $z=-2$. On the other hand $$|x+y+z|=|(x+y)+z| \geq ||x+y|-|z||$$ and by symmetry $$|x+y+z|\geq \max( ||x+y|-|z||, ||x+z|-|y||, ||y+z|-|x||).$$