Parallel coordinates in three dimensions and sharp spectral isoperimetric inequalities

Sharp Isoperimetric Inequalities via the Abp Method

We prove some old and new isoperimetric inequalities with the best constant using the ABP method applied to an appropriate linear Neumann problem. More precisely, we obtain a new family of sharp isoperimetric inequalities with weights (also called densities) in open convex cones of R. Our result applies to all nonnegative homogeneous weights satisfying a concavity condition in the cone. Remarka...

On Sharp Strichartz Inequalities in Low Dimensions

Recently Foschi gave a proof of a sharp Strichartz inequality in one and two dimensions. In this note, a new representation in terms of an orthogonal projection operator is obtained for the space time norm of solutions of the free Schrödinger equation in dimension one and two. As a consequence, the sharp Strichartz inequality follows from the elementary property that orthogonal projections do n...

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...