Geodesic
On one affine-invariant metric on the class of convex plane compacts
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 4, Tome 261 (1999), pp. 194-197

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Theorem. For every plane convex compacts $K_1,K_2\subset\mathbb R^2$ there exist an affine transformations $T_1$, $T_2$ such that $T_1(K_1)\subset K_2\subset T_2(K_1)$ and $S(T_2(K_1))111/16 S(T_1(K_1))$, where $S(K)$ means the square of a plane set $K\subset\mathbb R^2$. 
@article{ZNSL_1999_261_a13,
     author = {V. V. Makeev},
     title = {On one affine-invariant metric on the class of convex plane compacts},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {194--197},
     publisher = {mathdoc},
     volume = {261},
     year = {1999},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1999_261_a13/}
}
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V. V. Makeev. On one affine-invariant metric on the class of convex plane compacts. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 4, Tome 261 (1999), pp. 194-197. http://geodesic.mathdoc.fr/item/ZNSL_1999_261_a13/