# Tag Archives: karamata’s inequality

## Getting into Norms: a technical postscript

For mathematicians’ eyes only. The post doesn’t require much theoretical knowledge to understand, but I haven’t given many definitions.

In the last two posts I’ve been talking about an example I made with four-dimensional vectors $a, b, c$ such that $\|a\|_1 > \|b\|_1 > \|c\|_1$, $\|b\|_{2} > \|c\|_{2} > \|a\|_{2}$ and $\|c\|_{\infty}>\|a\|_{\infty}>\|b\|_{\infty}$. Finding it was more difficult than I at first expected, so I thought I would write the investigation up, which happily gives me an excuse to introduce a useful inequality.

My first thoughts were to choose something like $c=(10,0,0,0)$ and $a=(6,6,0,0)$ or $a=(4,4,4,0)$, and then I’d got stuck choosing $b$. So I decided to try to prove the opposite.

First of all, it’s not possible to create such an example in two dimensions. Continue reading