Prove that there is no norm in $\mathbb{R}^2$ such that the balls are six-pointed stars.
I'm aware this has to do with connexity, but not sure how to procced... If the star has empty interior then it's simple, but the problem isn't clear about that.
2026-03-25 10:55:33.1774436133
Inexistence of norm in $\mathbb{R}^2$
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If $\|v\|=\|w\|=r$, then $(\forall t\in[0,1]):\bigl\|tv+(1-t)w\bigr\|\leqslant r$. Let, let $v$ and $w$ be nearby vertices of a six-pointed star. Then the set $\{tv+(1-t)w\,|\,t\in(0,1)\}$ doesn't intersect the star. But I proved above that each of its elements has the same norm as $v$ and $w$.