Diameter of the set of midpoints of chords of a bounded closed set

The estimate is obtained for the diameter $d(S_n(a))$ of the set $S_n(a)$ of midpoints of chords of length $\ge a$ ($0<a\le1$) of a closed set of diameter 1 in the Euclidean space $E^n$, namely
$$ d(S_n(a))\leqslant\begin{cases} 1-a^2/2,&n=2, \\ \sqrt{1-a^2/2},&n\geqslant3, \end{cases} $$
and it is shown that the inequality cannot be improved.