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If $K \\subset {\\mathbb R}^2$ is a convex body, with $o \\in {\\text{int}}\\,K$, then $|K|\\cdot |K^*| \\ge 27/4$, with equality if and only if $K$ is a triangle and $o$ is its centroid. If $K \\subset {\\mathbb R}^2$ is a convex body, then we have $|K| \\cdot |[(K-K)/2)]^* | \\ge 6$, with equality if and only if $K$ is a triangle. 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Makai Jr., K. J. B\\\"or\\\"oczky, M. Meyer, S. Reisner","submitted_at":"2015-07-06T14:43:57Z","abstract_excerpt":"Let $K \\subset {\\mathbb R}^2$ be an $o$-symmetric convex body, and $K^*$ its polar body. Then we have $|K|\\cdot |K^*| \\ge 8$, with equality if and only if $K$ is a parallelogram. ($| \\cdot |$ denotes volume). If $K \\subset {\\mathbb R}^2$ is a convex body, with $o \\in {\\text{int}}\\,K$, then $|K|\\cdot |K^*| \\ge 27/4$, with equality if and only if $K$ is a triangle and $o$ is its centroid. If $K \\subset {\\mathbb R}^2$ is a convex body, then we have $|K| \\cdot |[(K-K)/2)]^* | \\ge 6$, with equality if and only if $K$ is a triangle. 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