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?
Scale-invariant geometries where $C/d \neq \pi$
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geometry
1 Answers
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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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0Thanks! Any other examples? – 2011-10-12
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1@Eelvex: with $l_\infty$, we again have $C/d=4$. The other $l_p$s all have $C/d$ constant but I haven't tried to work out what the values are. – 2011-10-12
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0any other examples apart from $L^p$ spaces? – 2011-10-13