Here is a general way to think about these kind of problems. The Minkowski theorem says that a polytope is uniquely determined by its normals and volumes of the facets. You can loose some of these conditions and ask for the optimum isoperimetric constantratio. In this case, in $\Bbb R^2$ you forget normals and conclude that inscribed polygon with given side lengths is optimal. A classical Lindelöf theorem does the opposite: in $\Bbb R^d$, it says that the optimal polytope with prescribed normals are circumscribed around the sphere (see e.g. here, Section 18.3).
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Here is a general way to think about these kind of problems. The Minkowski theorem says that a polytope is uniquely determined by its normals and volumes of facets. You can loose some of these conditions and ask for the optimum isoperimetric constant. In this case, in $\Bbb R^2$ you forget normals and conclude that inscribed polygon with given side lengths is optimal. A classical Lindelöf theorem does the opposite: in $\Bbb R^d$, it says that the optimal polytope with prescribed normals are circumscribed around the sphere (see e.g. here, Section 18.3). |
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