Same perimeter, different area.pic.twitter.com/A8hOQc2eRF
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Is it possible to deform a disk with a fixed perimeter in such a way that after the transformation the area is arbitrarily small?
Just degenerate an ellipse to a line, as @aknauft said.
I'm here to add: A Koch snowflake has infinite perimeter and finite area. If you deform a closed loop with fixed length into a (scaled down) Koch snowflake, the area will shrink each iteration. In the limit the area is zeropic.twitter.com/iOYAvEew3F
