Let X be a space and let S ⊂ X with a measure of set size |S| and boundary size |∂S|. Fix a set C ⊂ X called the constraining set. The constrained isoperimetric problem asks when we can find a subset S of C that maximizes the Følner ratio FR(S) = |S|/|∂S|. We consider different measures for subsets of R^2,R^3,Z^2,Z^3 and describe the properties that must be satisfied for sets S that maximize the Folner ratio. We give explicit examples.
College and Department
Physical and Mathematical Sciences; Mathematics
BYU ScholarsArchive Citation
Do, Minh Nhat Vo, "The Constrained Isoperimetric Problem" (2011). Theses and Dissertations. 2700.
amenability, isoperimetric, Folner ratio, cooling function, cooling field