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Approximation of convex bodies by axially symmetric bodies


Author: Marek Lassak
Journal: Proc. Amer. Math. Soc. 130 (2002), 3075-3084
MSC (1991): Primary 52A10, 52A27
DOI: https://doi.org/10.1090/S0002-9939-02-06404-3
Published electronically: March 14, 2002
Erratum: Proc. Amer. Math. Soc. 131 (2003), 2301.
MathSciNet review: 1908932
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Abstract: Let $C$ be an arbitrary planar convex body. We prove that $C$contains an axially symmetric convex body of area at least $\frac{2}{3}\vert C\vert$. Also approximation by some specific axially symmetric bodies is considered. In particular, we can inscribe a rhombus of area at least $\frac{1}{2}\vert C\vert$ in $C$, and we can circumscribe a homothetic rhombus of area at most $2\vert C\vert$ about $C$. The homothety ratio is at most $2$. Those factors $\frac{1}{2}$ and $2$, as well as the ratio $2$, cannot be improved.


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Additional Information

Marek Lassak
Affiliation: Instytut Matematyki i Fizyki ATR, 85-796 Bydgoszcz, Poland
Address at time of publication: Institut für Informatik, FU Berlin, D-14195, Berlin, Germany
Email: lassak@mail.atr.bydgoszcz.pl

DOI: https://doi.org/10.1090/S0002-9939-02-06404-3
Keywords: Convex body, axial symmetry, rhombus, isosceles triangle, area, approximation
Received by editor(s): March 1, 2000
Received by editor(s) in revised form: May 1, 2001
Published electronically: March 14, 2002
Additional Notes: This research was supported by Deutsche Forschungsgemeinschaft
Communicated by: Wolfgang Ziller
Article copyright: © Copyright 2002 American Mathematical Society

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