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**Camille91** Let $\displaystyle \mathbb{B}((x,y),r)$ on $\displaystyle \mathbb{R}^2$ be a ball in euclidean norm. We define a norm $\displaystyle ||x||$ that ball in this norm satisfes:

$\displaystyle B=\{ x \in \mathbb{R}^2: ||x||<1 \} = ( (-1,1) \times (-1,1) \cup \mathbb{B}((1,0),1) \cup \mathbb{B}((-1,0),1)$.

Is this norm determined by a dot product?

I think that this problem need to use parallelogram law ... but how?

Also we know that ball in eclidean norm is determined by the dot product. Maybe any draw should solve everything?