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This is an old result in convex geometry. See

E. Sas, ¨Uber ein Extremumeigenschaft der Ellipsen, Compositio Math. 6 (1939) 468– 470.

A. M. Macbeath, An extremal property of the hypersphere, Proc. Cambridge Philos. Soc. 47 (1951) 245–247.

Fedor Petrov's remark was right on target. Steiner symmetrization gives an easy proof (see Macbeath).

By the way, the equality case holds only for ellipses, which explains the titles of the papers above.

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This is an old result in convex geometry. See

E. Sas, ¨Uber ein Extremumeigenschaft der Ellipsen, Compositio Math. 6 (1939) 468– 470.

A. M. Macbeath, An extremal property of the hypersphere, Proc. Cambridge Philos. Soc. 47 (1951) 245–247.

Petrov's remark was right on target. Steiner symmetrization gives an easy proof (see Macbeath).