We show that the Banach–Mazur distance between the parallelogram and the affine-regular hexagon is \(\frac{3}{2}\) and we conclude that the diameter of the family of centrally-symmetric planar convex bodies is just \(\frac{3}{2}\). A proof of this fact does not seem to be published earlier. Asplund announced this without a proof in his paper proving that the Banach–Mazur distance of any planar centrally-symmetric bodies is at most \(\frac{3}{2}\). Analogously, we deal with the Banach–Mazur distances between the parallelogram and the remaining affine-regular even-gons.

中文翻译：

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从平行四边形到仿射-规则六边形和其他仿射-规则偶数子的Banach-Mazur距离

我们证明平行四边形与仿射正则六边形之间的Banach-Mazur距离为\（\ frac {3} {2} \），并得出结论，中心对称平面凸体族的直径仅为\（ \ frac {3} {2} \）。关于这一事实的证据似乎并未在较早之前发布。Asplund在他的论文中没有证明这一点便宣布了这一点，即证明任何平面的中心对称物体的Banach-Mazur距离最多为\（\ frac {3} {2} \）。类似地，我们处理平行四边形与其余仿射-正则偶角之间的Banach-Mazur距离。