In which $L^p$ metric is $\pi = 3.5$? I am interested because it's well known that $\pi$ can range from $3.14...$ to $4$ in $L^{\infty}$
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This $p$ probably cannot be expressed in any nice way. – Wojowu Oct 27 '17 at 08:08
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Isn't it $2\sqrt 2\approx 2.82$ in the $L^1$ norm? – Stefan Hante Oct 27 '17 at 08:12
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Woops. Sorry, I'll fix it. – mtheorylord Oct 27 '17 at 08:13
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There is a nice and helpful question exists: $\pi$ in arbitrary metric spaces. – m0nhawk Oct 27 '17 at 08:30
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Looks like @Wojowu is right - see here for the decidedly non-nice expression to solve – AakashM Oct 27 '17 at 08:30
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2@Wauzl I believe it isn't - the unit circle has circumference $2\sqrt{2}$ when measured using Euclidean distances, but taxicab-length of that circumference is $4$. – Wojowu Oct 27 '17 at 08:35
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1The fast search leads to $5.01797$ for $p$. Not sure if it's possible to have exact expression. – m0nhawk Oct 27 '17 at 08:35
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@Wojuwu: You seem to be right. This messes with my mind though :) – Stefan Hante Oct 27 '17 at 08:48
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Are you interested in a numerical approximation? Because i think it would be too hard to find a closed expression – supinf Oct 27 '17 at 09:25
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Not really, but out of curiosity how are they done? – mtheorylord Oct 27 '17 at 14:57