 Norm from a bounded symmetric convex neighbourhood

For a bounded convex open set $U$ containing zero and invariant under $x\mapsto -x$, its Minkowski functional is a norm. Inner and outer Euclidean balls give positive finite bounds; symmetry and scaling give absolute homogeneity; convexity gives the triangle inequality. Openness makes $U=\{p_U<1\}$, while boundedness prevents nonzero vectors from having zero gauge.