Geodesic
Diameter of the set of midpoints of chords of a bounded closed set
Matematičeskie zametki, Tome 6 (1969) no. 2, pp. 233-236 Cet article a éte moissonné depuis la source Math-Net.Ru

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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) 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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     author = {Yu. G. Dutkevich},
     title = {Diameter of the set of midpoints of chords of a~bounded closed set},
     journal = {Matemati\v{c}eskie zametki},
     pages = {233--236},
     year = {1969},
     volume = {6},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1969_6_2_a11/}
}
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Yu. G. Dutkevich. Diameter of the set of midpoints of chords of a bounded closed set. Matematičeskie zametki, Tome 6 (1969) no. 2, pp. 233-236. http://geodesic.mathdoc.fr/item/MZM_1969_6_2_a11/