Assume in 2-D $l_p$ space, The infinitesimal distance $ds=(|dx|^p+|dy|^p)^{1/p}$, and area $dA=dxdy$. What shape has the largest area/perimeter? My guess is the shape with largest area/perimeter in $l_p$ is $|x|^q+|y|^q=C$ where $1/p+1/q=1,p,q\geq1$.
I tested with numerical method using python, and indeed, my guess is true. But I would like to know why.