## Minimum Cuts and Maximum Area

- Date: 04/03/2007

Gilbert Strang (Massachusetts Institute of Technology)

Simon Fraser University

The oldest competition for an optimal shape (area-maximizing) was won by the circle. We want to give the thousandth proof !

Then we measure the perimeter in different ways, which changes the

problem (and has applications in medical imaging). If we use the line

integral of |dx| + |dy|, a square would win. Or if the boundary

integral of max(|dx|,|dy|) is given, a diamond has maximum area. When

the perimeter = integral of ||(dx,dy)|| around the boundary is given,

the area inside is maximized by a ball in the dual norm.

The second part describes the **max flow-min cut theorem** for

continuous flows. Usually it is for discrete flows on edges of graphs.

The maximum flow out of a region equals the capacity of the minimum

cut. This duality connects to the isoperimetric problems that produce

minimum cuts. But the flows are hard to find and a prize is unclaimed.

SFU CSC Distinguished Speaker Series 2007