We study the isoperimetric problem on $\mathbb{R}^1$ with a prescribed density function $f(x) = |x|$. Under these conditions, we find that isoperimetric $3$-bubble and $4$-bubble results satisfy a regular structure. As our regions increase in size, the intervals that form them alternate back-and-forth across the origin, with the smaller regions closer to the origin. This expands on previously known observations about the single- and double-bubble results on $\mathbb{R}$ with density $|x|^p$.
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 conditions: they have a point of zero density (at the origin), and they satisfy a "log-concavity" requirement $\left[ \log f \right]'' \leq 0$. Under these conditions, we find that isoperimetric $n$-bubbles satisfy a regular structure and can be identified for arbitrary $n$. This generalizes recent work done on the density function $|x|^p$, and stands in cont…
Klartag's needle decomposition technique enables one to obtain strong isoperimetric inequalities on Riemannian manifolds other than the classical known examples. As a result, in this paper, we obtain sharp isoperimetric inequalities for compact rank one symmetric spaces (CROSS). Namely, for the real projective space $\mathbb{R}P^n$, we demonstrate that the isoperimetric regions are given by either the geodesic balls or tubes around some $\mathbb{R}P^k\subset\mathbb{R}P^n$. For the complex projective space $\mathbb{C}P^n$, the isoperimetric regions are given by either the geodesic balls or tube…
In this paper we consider the isoperimetric profile of convex cylinders $K\times\mathbb{R}^q$, where $K$ is an $m$-dimensional convex body, and of cylindrically bounded convex sets, i.e, those with a relatively compact orthogonal projection over some hyperplane of $\mathbb{R}^{n+1}$, asymptotic to a right convex cylinder of the form $K\times\mathbb{R}$, with $K\subset\mathbb{R}^n$. Results concerning the concavity of the isoperimetric profile, existence of isoperimetric regions, and geometric descriptions of isoperimetric regions for small and large volumes are obtained.
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 they are stable (and $m > 0$). The implicit unjustified assumption is that the generating curve is convex.
Given a compact Riemannian manifold $M$ without boundary, we show that large isoperimetric regions in $M\times\mathbb{R}^k$ are tubular neighborhoods of $M\times\{x\}$, with $x\in\mathbb{R}^k$.
We study the isoperimetric problem for anisotropic left-invariant perimeter measures on $\mathbb R^3$, endowed with the Heisenberg group structure. The perimeter is associated with a left-invariant norm $ϕ$ on the horizontal distribution. We first prove a representation formula for the $ϕ$-perimeter of regular sets and, assuming some regularity on $ϕ$ and on its dual norm $ϕ^*$, we deduce a foliation property by sub-Finsler geodesics of $\mathrm C^2$-smooth surfaces with constant $ϕ$-curvature. We then prove that the characteristic set of $\mathrm C^2$-smooth surfaces that are locally extremal…
For a general family of graphs on $\mathbb{Z}^n$, we translate the edge-isoperimetric problem into a continuous isoperimetric problem in $\mathbb{R}^n$. We then solve the continuous isoperimetric problem using the Brunn-Minkowski inequality and Minkowski's theorem on Mixed Volumes. This translation allows us to conclude, under a reasonable assumption about the discrete problem, that the shapes of the optimal sets in the discrete problem approach the shape of the optimal set in the continuous problem as the size of the set grows. The solution is the zonotope defined as the Minkowski sum of the …
In this paper we solve several reverse isoperimetric problems in the class of $λ$-convex bodies, i.e., convex bodies whose curvature at each point of their boundary is bounded below by some $λ> 0$. We give an affirmative answer in $\mathbb{R}^3$ to a conjecture due to Borisenko which states that the $λ$-convex lens, i.e., the intersection of two balls of radius $1/λ$, is the unique minimizer of volume among all $λ$-convex bodies of given surface area. Also, we prove a reverse inradius inequality: in model spaces of constant curvature and arbitrary dimension, we show that the $λ$-convex lens (p…
The minimizers of the anisotropic fractional isoperimetric inequality with respect to the convex body $K$ in $\mathbb{R}^n$ are shown to be equivalent to star bodies whenever $K$ is strictly convex and unconditional. From this a Pólya-Szegö principle for anisotropic fractional seminorms is derived by using symmetrization with respect to star bodies.
We consider the problem of minimizing the relative perimeter under a volume constraint in an unbounded convex body $C\subset \mathbb{R}^{n+1}$, without assuming any further regularity on the boundary of $C$. Motivated by an example of an unbounded convex body with null isoperimetric profile, we introduce the concept of unbounded convex body with uniform geometry. We then provide a handy characterization of the uniform geometry property and, by exploiting the notion of asymptotic cylinder of $C$, we prove existence of isoperimetric regions in a generalized sense. By an approximation argument we…
We prove that finite perimeter subsets of $\mathbb{R}^{n+1}$ with small isoperimetric deficit have boundary Hausdorff-close to a sphere up to a subset of small measure. We also refine this closeness under some additional a priori integral curvature bounds. As an application, we answer a question raised by B. Colbois concerning the almost extremal hypersurfaces for Chavel's inequality.
We consider the class of $λ$-concave bodies in $\mathbb R^{n+1}$; that is, convex bodies with the property that each of their boundary points supports a tangent ball of radius $1/λ$ that lies locally (around the boundary point) inside the body. In this class we solve a reverse isoperimetric problem: we show that the convex hull of two balls of radius $1/λ$ (a sausage body) is a unique volume minimizer among all $λ$-concave bodies of given surface area. This is in a surprising contrast to the standard isoperimetric problem for which, as it is well-known, the unique maximizer is a ball. We solve…
Let $Ω$ be an open half-space or slab in $\mathbb{R}^{n+1}$ endowed with a perturbation of the Gaussian measure of the form $f(p):=\exp(ω(p)-c|p|^2)$, where $c>0$ and $ω$ is a smooth concave function depending only on the signed distance from the linear hyperplane parallel to $\partialΩ$. In this work we follow a variational approach to show that half-spaces perpendicular to $\partialΩ$ uniquely minimize the weighted perimeter in $Ω$ among sets enclosing the same weighted volume. The main ingredient of the proof is the characterization of half-spaces parallel or perpendicular to $\partialΩ$ as…
We consider an extremal problem for subsets of high-dimensional spheres that can be thought of as an extension of the classical isoperimetric problem on the sphere. Let $A$ be a subset of the $(m-1)$-dimensional sphere $\mathbb{S}^{m-1}$, and let $\mathbf{y}\in \mathbb{S}^{m-1}$ be a randomly chosen point on the sphere. What is the measure of the intersection of the $t$-neighborhood of the point $\mathbf{y}$ with the subset $A$? We show that with high probability this intersection is approximately as large as the intersection that would occur with high probability if $A$ were a spherical cap o…
We study upper bounds for the first non-zero eigenvalue of the Steklov problem defined on finite graphs with boundary. For finite graphs with boundary included in a Cayley graph associated to a group of polynomial growth, we give an upper bound for the first non-zero Steklov eigenvalue depending on the number of vertices of the graph and of its boundary. As a corollary, if the graph with boundary also satisfies a discrete isoperimetric inequality, we show that the first non-zero Steklov eigenvalue tends to zero as the number of vertices of the graph tends to infinity. This extends recent resul…
In this paper, we study the weighted Fermat-Frechet problem for a $\frac{N (N+1)}{2}-$tuple of positive real numbers determining $N$-simplexes in the $N$ dimensional $K$-Space ($N$-dimensional Euclidean space $\mathbb{R}^{N}$ if $K=0,$ the $N$-dimensional open hemisphere of radius $\frac{1}{\sqrt{K}}$ ($\mathbb{S}_{\frac{1}{\sqrt{K}}}^{N}$) if $K >0$ and the Lobachevsky space $\mathbb{H}_{K}^{N}$ of constant curvature $K$ if $K<0$). The (weighted) Fermat-Frechet problem is a new generalization of the (weighted) Fermat problem for $N$-simplexes. We control the number of solutions (weighted Ferm…
We develop criteria based on a calibration argument via discrete PDE and semidiscrete optimal transport, for finding sharp isoperimetric inequalities of the form $(\sharp Ω)^{d-1} \le C (\sharp \overrightarrow{\partialΩ})^d$ where $Ω$ is a subset of vertices of a graph and $\overrightarrow{\partialΩ}$ is the oriented edge-boundary of $Ω$, as well as the optimum isoperimetric shapes $Ω$. The method is a discrete counterpart to Optimal Transport and ABP method proofs valid in the continuum, and answers a question appearing in Hamamuki \cite{hamamuki}, extending that work valid for rectangular gr…
The $r$-parallel set of a measurable set $A \subseteq \mathbb R^d$ is the set of all points whose distance from $A$ is at most $r$. In this paper, we show that the surface area of an $r$-parallel set in $\mathbb R^d$ with volume at most $V$ is upper-bounded by $e^{Θ(d)}V/r$, whereas its Gaussian surface area is upper-bounded by $\max(e^{Θ(d)}, e^{Θ(d)}/r)$. We also derive a reverse form of the Brunn-Minkowski inequality for $r$-parallel sets, and as an aside a reverse entropy power inequality for Gaussian-smoothed random variables. We apply our results to two problems in theoretical machine le…
