= Norm determined by a convex radial unit ball
{title2=$p_B(x)=\inf\{t>0:x\in tB\}$}
Suppose $B$ contains zero, is <convex>, and meets every line $\mathbb Rv$, $v\ne0$, in the finite symmetric open segment $\{tv:|t|<\lambda(v)\}$ with $\lambda(v)>0$. Then its <Minkowski functional> is a <norm>, with $p_B(v)=1/\lambda(v)$ and $B=\{x:p_B(x)<1\}$. Absolute homogeneity follows by rescaling the segment. If $s>p_B(x)$ and $t>p_B(y)$, convexity puts $(x+y)/(s+t)$ in $B$, giving $p_B(x+y)<s+t$; passing to the infima proves the triangle inequality. Conversely the open unit ball uniquely determines any norm by the displayed formula. No prior topology on the vector space is needed for these radial hypotheses.
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