Massachusetts Institute of Technology

Maximum Area and Minimum Cuts with New Measures of Perimeter

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 R2, 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 l2 norm, that ball is a circle. Our proof comes directly from the calculus of variations.
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. The problem arises in medical imaging.

Wednesday, February 7, 2007
Amos Eaton 214, 4:00 p.m.