Abstract:
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.
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