## Symmetrization

Write-up of a departmental faculty seminar, October 2, 2009.

Solutions to problems in geometry and physics and even in the social sciences tend to be symmetric. As prime example, the solution to the isoperimetric problem, which seeks the least-perimeter way to enclose given volume in **R**^{3}, is a sphere, the most symmetric of all shapes. One way to prove this is to show that anything else improves as you make it more symmetric. For thousands of years, mathematicians have been looking for good ways to make shapes more symmetric and to prove that as they get more symmetric they “get better,” for example, enclose the same volume with less perimeter.

My favorite references are Burago and Zalgaller [BZ, §9.2] and Ros [R1, §3.2]. This talk is based on [MHH]. Gromov [G, §9.4] provides some sweeping remarks and generalizations, including most of our results.

**1. Steiner symmetrization **[St, 1838] replaces every vertical slice of a region in **R**^{3} with a centered interval of the same length, as in Figure 1. By calculus, the volume does not change, but one can show that the perimeter decreases (or remains the same).

Figure 1. Steiner symmetrization replaces every vertical slice with a centered interval of the same length. www.math.utah.edu/~treiberg/Lect.html