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Is there a geometry where everywhere, or locally: $ \frac{C}{d} = \mathrm{constant} \neq \pi$ $C, d$ being the circumference and diameter of a circle?

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Yes. For example, if you equip $\mathbb{R}^2$ with the $l_1$ metric, then a circle of diameter $d$ has circumference $4d$.

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    any other examples apart from $L^p$ spaces?2011-10-13