# Approximation of Convex Bodies by Rhombi and by other Axially
Symmetric Bodies

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Marek Lassak

Institut für Informatik

Freie Universität Berlin

Takustr. 9, D-14195 Berlin

email: lassak@inf.fu-berlin.de

Report B 99-22

December 1999

**Abstract:**
Let *C* be an arbitrary planar convex body. We prove that *C*
contains an axially symmetric convex body of area at least 2/3 |* C* |. Also
approximation by some specific axially symmetric bodies is considered. In
particular, we can inscribe a rhombus of area at least ½ |* C* | in
*C*, and we can circumscribe a homothetic rhombus of area at most 2 | *C * | about *C*. The homothetic ratio is at most 2. Those coefficients ½ and 2 and also the homothetic ratio 2 cannot be improved.

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