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Maximum area and minimum cuts with new measures of perimeter
|Speaker: ||Gilbert Strang, MIT|
|Location: ||1147 MSB|
|Start time: ||Mon, Feb 26 2007, 4:10PM|
The oldest competition for an optimal shape (area-maximizing) was won by the circle. But if the fixed perimeter is measured by 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.
For any norm in R^2, we show that when the integral of ||(dx,dy)|| around the boundary is prescribed, the area inside is maximized by a ball in the dual norm. When ||.|| is the l^2 norm, that ball is a circle. Our proof comes directly from the calculus of variations, where Busemann's original proof used inequalities from convex geometry. This isoperimetric problem has application to computing minimum cuts and maximum flows in a plane domain. There the key is Cheeger's isoperimetric problem with shape inside a fixed set.