Let $S$ be a bounded convex subset of the plane of unit area. Can we guarantee the existence of a centrally-symmetric subset $C⊆ S$ of area $2/3$?
If $S$ is any triangle, this bound is tight, attained by a hexagon whose vertices are $1/3$ of the way across each side:
I suspect the triangle is the unique worst case for this problem, but didn't see a great way to prove it - perhaps there is some other convex shape which only has a centrally symmetric subset of area at most $0.6$ or something, though I would be quite surprised.
It's easy to find a $C$ with at least half the area of $S$, because every convex set contains a rectangle of at least half its area (see e.g. this MO thread for references). How much can we improve this lower bound?
In the event of a positive resolution, I am curious whether one can find such a subset only using hexagonal $C$.
I imagine this question is discussed somewhere in the literature; if so, pointers to relevant papers or extensions to higher dimensions would be welcome.
Yes, this is the Kovner-Besikovich Theorem.