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Consider a closed curve on the plane so that if we perform any translation and dilation on it, the resulting curve intersects the original curve at most twice. Does this property characterize the closed curves that bound strictly convex regions on the plane?

This was a "doodling while waiting for the bus" problem this morning.

I think I have a sketch of a proof in one direction: if we expand a closed curve large enough, some part of the curve will look like a straight line: in fact, you can approximate a straight line of any slope this way. Since curves that aren't strictly convex will intersect some straight line more than twice, this property cannot hold for curves that don't bound strictly convex regions.

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Hint: Strictly convex function has at most two zeros. –  Anton Petrunin Jan 25 '12 at 3:53
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