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Mat. Zametki, 1969, Volume 6, Issue 2, Pages 233–236 (Mi mz6927)  

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

Yu. G. Dutkevich

Leningrad State University named after A. A. Zhdanov

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.

Full text: PDF file (278 kB)

English version:
Mathematical Notes, 1969, 6:2, 593–595

Bibliographic databases:

UDC: 513.78
Received: 16.09.1968

Citation: Yu. G. Dutkevich, “Diameter of the set of midpoints of chords of a bounded closed set”, Mat. Zametki, 6:2 (1969), 233–236; Math. Notes, 6:2 (1969), 593–595

Citation in format AMSBIB
\Bibitem{Dut69}
\by Yu.~G.~Dutkevich
\paper Diameter of the set of midpoints of chords of a~bounded closed set
\jour Mat. Zametki
\yr 1969
\vol 6
\issue 2
\pages 233--236
\mathnet{http://mi.mathnet.ru/mz6927}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=254737}
\zmath{https://zbmath.org/?q=an:0188.55101|0183.26903}
\transl
\jour Math. Notes
\yr 1969
\vol 6
\issue 2
\pages 593--595
\crossref{https://doi.org/10.1007/BF01093705}


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  • Математические заметки Mathematical Notes
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