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In his remarkable book On the Sphere and Cylinder, where he came tantalizingly close to discovering calculus, Archimedes showed that the area of the portion of the sphere contained between a pair of ...

Let $K\subset R^2$ be a convex body, i.e., a compact convex set with interior points. A plank $P$ is the region between a pair of parallel lines in $R^2$. Let us say that $P$ intersects $K$ properly ...