7,053 research outputs found
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 with density
We prove the existence of isoperimetric regions in 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 by using symmetrization techniques.Comment: 19 pages, 3 figure
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