Approximation of convex bodies by axially symmetric bodies

Lassak, Marek

doi:10.1090/S0002-9939-02-06404-3

Approximation of convex bodies by axially symmetric bodies
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Proc. Amer. Math. Soc. 130 (2002), 3075-3084 Request permission

Erratum: Proc. Amer. Math. Soc. 131 (2003), 2301-2301.

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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