= Solution
The interior of the diamond is $B=\{(x_1,x_2):|x_1|+|x_2|<1\}$. Its <Minkowski functional>, or the uniqueness argument already proved, gives the <L1 norm>:
$$
\boxed{\|(x_1,x_2)\|_B=|x_1|+|x_2|.}
$$
The final unheaded set $C$ cannot be an <open unit ball> of a <norm>, because every norm ball is <convex> by homogeneity and the <triangle inequality>. Both $p=(9/10,0)$ and $q=(0,9/10)$ lie in $C$: their squared-distance expression is $101/100>1$. Their midpoint has that expression
$$
2(1-9/20)^2=121/200<1,
$$
so $(p+q)/2\notin C$. Thus $\boxed{\text{no norm has }C\text{ as its open unit ball}}$. The strict inequalities in the definition are respected by this explicit counterexample.
Back to article page
