Exact Face-isoperimetric Inequalities

Lp AFFINE ISOPERIMETRIC INEQUALITIES

Affine isoperimetric inequalities compare functionals, associated with convex (or more general) bodies, whose ratios are invariant under GL(n)-transformations of the bodies. These isoperimetric inequalities are more powerful than their better-known relatives of a Euclidean flavor. To be a bit more specific, this article deals with inequalities for centroid and projection bodies. Centroid bodies...

Isoperimetric Inequalities and Eigenvalues

Isoperimetric Inequalities in Crystallography

The study of the isoperimetric problem in the presence of crystallographic symmetries is an interesting unsolved question in classical differential geometry: Given a space group G, we want to describe, among surfaces dividing Euclidean 3-space into two G-invariant regions with prescribed volume fractions, those which have the least area per unit cell of the group. We know that this periodic iso...

Isoperimetric Inequalities and Eigenvalues

An upper bound is given on the minimum distance between i subsets of the same size of a regular graph in terms of the i-th largest eigenvalue in absolute value. This yields a bound on the diameter in terms of the i-th largest eigenvalue, for any integer i. Our bounds are shown to be asymptotically tight. A recent result by Quenell relating the diameter, the second eigenvalue, and the girth of a...

Discrete Isoperimetric Inequalities

One of the earliest problems in geometry is the isoperimetric problem, which was considered by the ancient Greeks. The problem is to find, among all closed curves of a given length, the one which encloses the maximum area. Isoperimetric problems for the discrete domain are in the same spirit but with different complexity. A basic model for communication and computational networks is a graph G =...