Characterizations of Steiner points

To each convex compact $A$ in Euclidian space $E^n$ there corresponds a point $S(A)$ from $E^n$ such that 1) $S(x)=x$ for $x\in E^n$, 2) $S(A+B)=S(A)+S(B)$, 3) $S(A_i)\to0$, if $A_i$ converges in the Hausdorff metric to the unit sphere in $E^n$, then $S(A)$ is the Steiner point of the set $A$. The theorem improves certain earlier results on characterizations of the Steiner point.