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Proceedings of the American Mathematical Society

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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
Published electronically: March 14, 2002
Erratum: Proc. Amer. Math. Soc. 131 (2003), 2301-2301.
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 $\frac {2}{3}|C|$. Also approximation by some specific axially symmetric bodies is considered. In particular, we can inscribe a rhombus of area at least $\frac {1}{2}|C|$ in $C$, and we can circumscribe a homothetic rhombus of area at most $2|C|$ 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

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