Problem of the Week 1210
A simple closed curve C in the plane is called "shrinkable" if all shrunk versions of C can fit into the closure of the interior of C.
Any convex curve is shrinkable.
Any star-shaped curve is shrinkable.
Find an example of a shrinkable curve that is not star-shaped.
A shrunk version of C is the curve obtained by choosing some shrinking factor p between 0 and 1 and using the points (px, py), where (x, y) is on C. The problem is about all shrinkings, that is, all possible choices of the shrinking factor.
The fitting can use any translation and rotation.
We allow the shrunk curve to fit inside the set consisting of the interior of C and C itself; i.e., it need not be strictly inside C.
A curve is "star-shaped" if there is a point in its interior such that the straight line from the point to any point on the curve lies inside the curve.
The "Drink Me" potion in Wonderland caused Alice to shrink to a fraction of her normal size.
Source: Dan Asimov formulated and solved this problem in 1966.
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