2
$\begingroup$

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?

1 Answers 1

3

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

  • 0
    Thanks! Any other examples?2011-10-12
  • 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
  • 0
    any other examples apart from $L^p$ spaces?2011-10-13