Related papers: The Log Convex Density Conjecture in Hyperbolic Sp…
The isoperimetric problem with a density or weighting seeks to enclose prescribed weighted volume with minimum weighted perimeter. According to Chambers' recent proof of the log-convex density conjecture, for many densities on…
We prove that geodesic balls centered at some base point are isoperimetric in the real hyperbolic space $H_{\mathbb R}^n$ endowed with a smooth, radial, strictly log-convex density on the volume and perimeter. This is an analogue of the…
We prove a counterpart of the log-convex density conjecture in the hyperbolic plane.
We completely characterize isoperimetric regions in R^n with density e^h, where h is convex, smooth, and radially symmetric. In particular, balls around the origin constitute isoperimetric regions of any given volume, proving the Log-Convex…
We examine the vertical component of surface area in the warped product of a Euclidean interval and a fiber manifold with product density. We determine general conditions under which vertical fibers minimize vertical surface area among…
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…
We study the isoperimetric problem on $\mathbb{R}^1$ with a prescribed density function $f$ that affects how area and perimeter are measured. We examine density functions that are symmetric, radially increasing, and satisfy two additional…
In this paper, we prove a class of weighted isoperimetric inequalities for bounded domains in hyperbolic space by using the isoperimetric inequality with log-convex density in Euclidean space. As a consequence, we remove the horo-convex…
In this paper, we consider the isoperimetric problem in the space $\mathbb{R}^N$ with density. Our result states that, if the density f is l.s.c. and converges to a positive limit at infinity, being smaller than this limit far from the…
In this paper we consider the isoperimetric problem with double density in an Euclidean space, that is, we study the minimisation of the perimeter among subsets of $\mathbb{R}^n$ with fixed volume, where volume and perimeter are relative to…
The classic double bubble theorem says that the least-perimeter way to enclose and separate two prescribed volumes in $\mathbb{R}^N$ is the standard double bubble. We seek the optimal double bubble in $\mathbb{R}^N$ with density, which we…
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.
We consider the problem of lower bounding a generalized Minkowski measure of subsets of a convex body with a log-concave probability measure, conditioned on the set size. A bound is given in terms of diameter and set size, which is sharp…
In hyperbolic space density cannot be defined by a limit as we define it in Euclidean space. We describe the local density bounds for sphere packings and we discuss the different attempts to define optimal arrangements in hyperbolic space.
The convex body isoperimetric conjecture in the plane asserts that the least perimeter to enclose given area inside a unit disk is greater than inside any other convex set of area $\pi$. In this note we confirm two cases of the conjecture:…
The isoperimetric problem is one of the oldest in geometry and it consists of finding a surface of minimum area that encloses a given volume $V$. It is particularly important in physics because of its strong relation with stability, and…
One of the basic problems in discrete geometry is to determine the most efficient packing of congruent replicas of a given convex set $K$ in the plane or in space. The most commonly used measure of efficiency is density. Several types of…
We have discovered a "little" gap in our proof of the sharp conjecture that in $\mathbb{R}^n$ with volume and perimeter densities $r^m$ and $r^k$, balls about the origin are uniquely isoperimetric if $0 < m \leq k - k/(n+k-1)$, that is, if…
In this paper we consider the problem of minimizing the relative perimeter under a volume constraint in the interior of a conically bounded convex set, i.e., an unbounded convex body admitting an \emph{exterior} asymptotic cone. Results…
In previous work a probabilistic approach to controlling difficulties of density in hyperbolic space led to a workable notion of optimal density for packings of bodies. In this paper we extend an ergodic theorem of Nevo to provide an…