Geodesic
A kinematic formula for affine diameters and affine medians of a convex set
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 7, Tome 280 (2001), pp. 234-238 Cet article a éte moissonné depuis la source Math-Net.Ru

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For a planar convex set $K$ with $C^2$-smooth boundary, the area of the set of the points lying on a given number of affine diameters of $K$ is estimated. As a corollary, it is proved that the area of $K$ is at most $\pi M^2/4$, where $M$ is the largest length of a chord of $K$ halving the area of $K$. 
@article{ZNSL_2001_280_a16,
     author = {V. V. Makeev},
     title = {A kinematic formula for affine diameters and affine medians of a convex set},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {234--238},
     year = {2001},
     volume = {280},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2001_280_a16/}
}
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V. V. Makeev. A kinematic formula for affine diameters and affine medians of a convex set. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 7, Tome 280 (2001), pp. 234-238. http://geodesic.mathdoc.fr/item/ZNSL_2001_280_a16/