In polytopes, small balls about some vertex minimize perimeter

In (the surface of) a convex polytope P^n in R^n+1, for small prescribed volume, geodesic balls about some vertex minimize perimeter. This revision corrects a mistake in the mass bound argument in the proof of Theorem 3.8.Comment: Revision corrects a mistake in the mass bound argument in the proof of Theorem 3.8. J. Geom. Anal., to appea

The Space of Planar Soap Bubble Clusters

Soap bubbles and foams have been extensively studied by scientists, engineers, and mathematicians as models for organisms and materials, with applications ranging from extinguishing fires to mining to baking bread. Here we provide some basic results on the space of planar clusters of n bubbles of fixed topology. We show for example that such a space of clusters with positive second variation is an n-dimensional manifold, although the larger space without the positive second variation assumption can have singularities. Earlier work of Moukarzel showed how to realize a cluster as a generalized Voronoi partition, though not canonically.Comment: 10 pages, 3 figure

Hexagonal surfaces of Kapouleas

For the "hexagonal" norm on R^3, for which the isoperimetric shape is a hexagonal prism rather than a round ball, we give analogs of the compact immersed constant-mean-curvature surfaces of Kapouleas.Comment: 11 page

Existence of isoperimetric regions in Rn\R^n with density

We prove the existence of isoperimetric regions in $\R^n$ with density under various hypotheses on the growth of the density. Along the way we prove results on the boundedness of isoperimetric regions.Comment: 31 pages, 4 figure

Planar Clusters

We provide upper and lower bounds on the least-perimeter way to enclose and separate n regions of equal area in the plane. Along the way, inside the hexagonal honeycomb, we provide minimizers for each n .Comment: 13 page

Steiner and Schwarz symmetrization in warped products and fiber bundles with density

We provide very general symmetrization theorems in arbitrary dimension and codimension, in products, warped products, and certain fiber bundles such as lens spaces, including Steiner, Schwarz, and spherical symmetrization and admitting density.Comment: 9 page

The Isoperimetric Problem in Higher Codimension

We consider three generalizations of the isoperimetric problem to higher codimension and provide results on equilibrium, stability, and minimization.Comment: 13 pages, 1 figure; v2: Minor revision to appear in Manuscripta Mathematic

When Soap Bubbles Collide

Can you fill R^n with a froth of "soap bubbles" that meet at most n at a time? Not if they have bounded diameter, as follows from Lebesgue's Covering Theorem. We provide some related results and conjectures.Comment: 9 pages, 5 figures, better proof of Prop 2.4. To appear in Amer. Math. Monthl

Proof of the Double Bubble Conjecture

We prove that the standard double bubble provides the least-area way to enclose and separate two regions of prescribed volume in \Bbb R^3.Comment: 31 pages, published versio

On the isoperimetric problem in Euclidean space with density

We study the isoperimetric problem for Euclidean space endowed with a continuous density. In dimension one, we characterize isoperimetric regions for a unimodal density. In higher dimensions, we prove existence results and we derive stability conditions, which lead to the conjecture that for a radial log-convex density, balls about the origin are isoperimetric regions. Finally, we prove this conjecture and the uniqueness of minimizers for the density $\exp (|x|^2)$ by using symmetrization techniques.Comment: 19 pages, 3 figure