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Mike66: Mother's Pie

Mike66A circular pie may only be cut (and eaten) in radial "wedges." How much pie must be eaten before the remainder can be put on a smaller plate?

For example:

Equivalently, the problem is to give bounds on the maximal total area of closed sectors of unit radius which can be placed disjointly within an open unit circle.

You are invited to send me (Mike dot Paterson at dcs dot warwick dot ac dot uk) solutions before September 13, 2008. These could be lower bounds, upper bounds or both, according to taste.

(This problem is a tribute to my mother.)

A pdf file with the problem is also available.

Some progress with solutions is reported here. A generalisation of the problem is also introduced.