Questions tagged [isoperimetric-problems]
The isoperimetric-problems tag has no usage guidance.
95
questions
34
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5
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Why is Fourier analysis so handy for proving the isoperimetric inequality?
I have just completed an introductory course on analysis, and have been looking over my notes for the year. For me, although it was certainly not the most powerful or important theorem which we ...
31
votes
1
answer
2k
views
A long-lasting conjecture of Pólya & Szegő
There is a conjecture by Pólya & Szegő (~1950, stated in p. 159 of their book Isoperimetric Inequalties in Mathematical Physics) which is as follows:
"Of all $n$-gons of a fixed area, the regular ...
24
votes
3
answers
1k
views
Isoperimetric inequality on a Riemannian sphere
Consider a two-dimensional sphere with a Riemannian metric of total area $4\pi$. Does there exist a subset whose area equals $2\pi$ and whose boundary has length no greater than $2\pi$?
(To avoid ...
22
votes
0
answers
502
views
Sphere with bounded curvature
Let $V$ be a body in $\mathbb{R}^3$ bounded by a smooth sphere with principal curvatures at most 1 (by absolute value).
Is it true that
$$\mathop{\rm vol} V\ge \mathop{\rm vol} B,$$
where $B$ denotes ...
20
votes
2
answers
2k
views
"a shape that ... lies halfway between a square and a circle"
An article in the
Notices of the AMS, Volume 61, Issue 10, 2014
(PDF download link),
on Khot's Unique Games Conjecture, says this:
Another group ... found a
shape that in a certain sense lies ...
17
votes
2
answers
945
views
Isoperimetric-like inequality for non-connected sets
The classical isoperimetric inequality can be stated as follows: if $A$ and $B$ are sets in the plane with the same area, and if $B$ is a disk, then the perimeter of $A$ is larger than the perimeter ...
16
votes
1
answer
1k
views
Optimal 8-vertex isoperimetric polyhedron?
I know from Marcel Berger's
Geometry Revealed:
A Jacob's Ladder to Modern Higher Geometry
(p.531)
that it is not yet established which polyhedron in $\mathbb{R}^3$ on 8 vertices achieves the optimal ...
16
votes
0
answers
588
views
Isoperimetric inequality and geometric measure theory
The following version of the isoperimetric inequality can be easily deduced from the Brunn-Minkowski inequality:
Theorem. If $K\subset\mathbb{R}^n$ is compact, then $$ |K|^{\frac{n-1}{n}}\leq n^{-1}...
15
votes
3
answers
1k
views
Stronger version of the isoperimetric inequality
I have been searching for a version of the isoperimetric inequality which is something like:
$P(\Omega) - 2\sqrt{\pi} Vol(\Omega)^{1/2} \geq \pi (r_{out}^2 - r_{in}^2)$ where $r_{out}$ and $r_{in}$ ...
12
votes
1
answer
656
views
General Isoperimetric Inequality via Representation Theory of SO(n)
Is there a known proof of the $n$-dimensional isoperimetric inequality which generalizes Hurwitz's proof using Fourier analysis in the $2$-dimensional case?
Specifically, I imagine such a proof would ...
12
votes
0
answers
349
views
A variation on the local Günther inequality
This question is about a variation on the Günther (also known as Günther-Bishop) inequality for manifolds of sectional curvature bounded from above. With Greg Kuperberg, we would deduce from it a ...
11
votes
2
answers
2k
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Levy's isoperimetric inequality for sphere
Let me recall subj:
If $s>0$, $A$ and $B$ are two subsets of $\mathbb{S}^{n}$, $|A|=|B|$ ($|\cdot|$ stands for the Lebesgue measure on the sphere) and $B$ is a cup $B=\{ (x_1,x_2,\dots,x_n)\in \...
11
votes
2
answers
739
views
Isoperimetric inequality in complex hyperbolic space
Let $\mathbb{H}_\mathbb{C}^n$ be n-dimensional complex hyperbolic space.
This space is a complex analog of hyperbolic space. It is isometric to the quotient of hyperboloid
$$|z_0|^2-|z_1|^2-\dots-|...
10
votes
4
answers
620
views
Packing and isoperimetrics
Suppose a manufacturer bottles small units of liquid and ships them via very large trucks.
If the transportation cost nothing, spherical bottles would minimize the packaging cost (isoperimetric ...
10
votes
2
answers
886
views
Is there a volume-preserving diffeomorphism of the disk with prescribed singular values?
This is a cross-post. While working on a variational problem, I have reached to the following question.
Let $0<\sigma_1<\sigma_2$ satisfy $\sigma_1\sigma_2=1$, and let $D \subseteq \mathbb{R}^2$...
10
votes
1
answer
445
views
Isoperimetric inequality for closed curves in $\mathbb{R}^n$
A well known isoperimetric inequality for closed curves in $\mathbb{R}^2$ can be generalized to closed curves in $\mathbb{R}^{2n}$, see: https://mathoverflow.net/a/321505/121665.
I have two questions:
...
9
votes
1
answer
2k
views
Isoperimetry and Poincaré Inequality
What are the known relations between isoperimetric and Poincaré inequalities on manifolds?
For example, for manifolds with a lower bound on Ricci curvature, the Cheeger-Buser inequality relates the ...
9
votes
1
answer
544
views
What is the shape of the $n$-gon which gives the maximum of a function?
What is the shape of the $n$-gon $P_1P_2\cdots P_n$ which gives the maximum of $A_n$? The quantity $A_n$ is defined by
$$ A_n = \frac{{\sum_{i\lt{j}\le{n}}{\lvert P_i P_j\rvert}^2}-{\sum_{i=1}^{n}{\...
8
votes
3
answers
285
views
Set with small internal radius, small perimeter and prescribed area
Given a regular set $E\subset \mathbb R^2$ define
$$
R(E) = \sup\{r\colon \exists x,\ B(x,r)\subseteq E\}
$$
to be the radius of the largest circle contained in $E$ and let $|\partial E|$ be the ...
8
votes
1
answer
263
views
Isoperimetric inequality on the plane
Let $A$ be a connected compact domain with smooth boundary in the Euclidean 2-plane. Assume its diameter is at most $d$. Assume that the second fundamental form of the boundary is at most $-c$ where $...
8
votes
2
answers
779
views
A hypercube-related graph
For integer $n\ge 3$, consider the graph on the set of all even vertices of the $n$-dimensional hypercube $\{0,1\}^n$ in which two vertices are adjacent whenever they differ in exactly two coordinates....
8
votes
1
answer
438
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An isoperimetric-type inequality inside a cube
I am looking for a reference for the following inequality: if $\Omega \subset [0,1]^d$ satisfies $\mbox{vol}(\Omega) \leq 1/2$, then
$$ \mathcal{H}^{d-1}\left( \partial\Omega \cap (0,1)^d\right) \geq ...
8
votes
1
answer
234
views
Poincare's argument for maximizing the Coulomb energy
For $\Omega\subset \mathbb{R}^3$ a region with $|\Omega| = |B_1|$, let
$$
C(\Omega) = \int_\Omega\int_\Omega \frac{dxdy}{|x-y|}
$$
denote the Coulomb (or gravitational, etc) energy.
Poincaré is ...
8
votes
1
answer
318
views
Mass transportation proof of the Gaussian isoperimetric inequality?
In his book "Topics in optimal transportation", Graduate Studies in Mathematics 58, AMS 2003,
Villani presents a proof, due to Gromov, of the classical isoperimetric inequality
in Euclidean ...
8
votes
0
answers
109
views
Hölder isoperimetric problem
Denote by $S_r$ the usual circle of radius $r$, with the path metric ($d(x,y) = r\theta$, where $\theta$ is the angle between the vectors $x$ and $y$), and let $\alpha \in (1/2,1)$. Consider the ...
7
votes
2
answers
356
views
Closed curve whose neighborhood is as large as possible
Let $C$ be a closed curve in the plane and let $N_\epsilon(C)$ be an $\epsilon$-neighborhood of $C$, like this:
(ignore the fact that the "curve" is polygonal in this picture, I drew it in MATLAB)
...
7
votes
2
answers
404
views
Graph which do not satisfy a pseudo-Poincaré inequality
Say that an infinite (connected) graph (with vertices of bounded degree) satisfies a $\ell_1$-pseudo-Poincaré inequality if there is a constant $C>0$ so that for any $n \in \mathbb{N}$ for any ...
7
votes
2
answers
506
views
Visual proof of convergence for Steiner's symmetrization
I want to find a visual proof of the following fact:
For any convex figure in the plane there is a sequence of Steiner's symmetrizations that makes it arbitrary close to a circular disc.
All ...
7
votes
3
answers
613
views
When is perimeter continuous under Hausdorff convergence?
It is known that the perimeter is lower semicontinuous for the convergence of sets. Two variants are widely known:
(Golab's theorem) in $\Bbb{R}^2$ if the sets $\Omega_n$ converge to $\Omega$ in the ...
7
votes
1
answer
207
views
Isoperimetric type inequality in $\mathbb{R}^2$
Fix $L \in (0,\infty)$ and consider $\mathcal{C}_L$ defined as follows:
\begin{align*}
\mathcal{C}_L := \{ \gamma:[0,1] \rightarrow \mathbb{R}^2 |~ \gamma \text{ is smooth and length($\gamma$)$=L$ }\}....
7
votes
1
answer
281
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Convexity of Isoperimetric Domains
I am interested in what is known about the convexity of isoperimetric domains in compact Cartan-Hadamard manifolds (Riemannian manifolds that are complete and simply-connected and have non-positive ...
7
votes
1
answer
424
views
An isoperimetric type of inequality in terms of Wasserstein distance/Optimal transport
Let $A \subset \mathbb{R}^n$ be a region having the same volume as an $n$ dimensional ball $B^n_R$ with radius $R$ centring at the origin.
Isoperimetric inequality says:
$ Vol_{n-1} \partial A \geq ...
7
votes
0
answers
648
views
Least area minimal hypersurface of $\mathbb C P^{n+1}$
After a few lectures on min-max for minimal hypersurfaces and isoperimetric problems, and seeing in several instances that the least area minimal hypersurface of the round sphere is an equator, I was ...
6
votes
2
answers
424
views
$L^{p}$ isoperimetric inequalities on the Hamming cube
Let $A \subset \{-1,1\}^{n}$ be a subset of the Hamming cube with cardinality $|A|=2^{n-1}$. Define $w_{A} : \{-1,1\}^{n} \to \mathbb{N}\cup \{0\}$ so that $w_{A}(x)$ to be number of boundary edges ...
6
votes
0
answers
176
views
Optimal planar net for catching convex shapes
Imagine you want to make a net out of string to filter and catch objects of
a certain size, minimizing the length of string employed.
(This actually arises in filtering biological impurities from ...
5
votes
3
answers
349
views
Perimeter/Neighborhood of a graph on grid
Hello,
I have a $\sqrt{n}\times\sqrt{n}$ lattice graph $G=(V,E)$ i.e. vertices on said 2-dim integer lattice, and two vertices have an edge if their $L_1$ distance is one.
Now I want to claim ...
5
votes
1
answer
183
views
An Indepth Look at Isoperimetry in the Cayley Graph Generated by All Transpositions
Let $\Omega_n$ denote the symmetric/permutation group on $n$ objects.
Let $T_n \subseteq \Omega_n$ denote the set of transpositions.
Drop the $n$-subscripts.
Define the Cayley graph $G = (\Omega, E)$ ...
5
votes
1
answer
174
views
Area of a elliptic surface confined by a sphere
Let $S$ be a surface enclosed inside the unit sphere in $R^3$. If every point of S is elliptic, then must $\operatorname{Area}(S)≤\operatorname{Area}(S^2)$?
5
votes
1
answer
303
views
Panning for gold nuggets: a type of isoperimetric problem
Let $C$ be a unit-radius circle in the plane.
Suppose you have a total length $L$ of string available, and
your task is to connect chords of $C$ using no more
than $L$ of string to minimize the ...
5
votes
1
answer
627
views
An inequality inspired by the isoperimetric inequality
Let us consider the simplest isoperimetric inequality. Consider a smooth simple closed curve given by $r=\rho(\theta)$ in polar coordinates, where $\rho(\theta)>0$ can be regarded as a smooth ...
5
votes
1
answer
175
views
"Isoperimetric inequality" for self intersecting closed surfaces?
As the title suggests, I am trying to find something like an isoperimetric inequality for smooth immersions of the 2-sphere into $\mathbb{R}^3$ that relates the surface area to the enclosed 3d-volume. ...
5
votes
0
answers
72
views
Dimension reduction and isoperimetric inequality
$\newcommand{\II}{\mathit{II}}$The isoperimetric inequality $\II_n$ in ${\mathbb R}^n$ is
$$\frac{{\rm vol}_nU}{{\rm vol}_nB_n}\le\left(\frac{{\rm vol}_{n-1}\partial U}{{\rm vol}_{n-1}\partial B_n}\...
5
votes
0
answers
84
views
Isoperimetric profile with obstacle
Fix two open smooth bounded domains $\Omega_-$ and $\Omega_+$ with $\overline{\Omega}_-\subset \Omega_+$ in a complete Riemannian manifold ($\mathbb{R}^n$ is already interesting to me).
I was ...
4
votes
2
answers
2k
views
Isoperimetric inequality in negative sectional curvature
Let $M$ be a complete, non-compact, simply connected Riemannian manifold of dimension $n$ whose sectional curvatures are bounded above by $\kappa<0$. I want to prove that for any open subset $\...
4
votes
2
answers
757
views
The Isoperimetric problem for domains constrained to lie between two parallel planes
It is well known that for a given volume $V$, a sphere is the shape that minimizes the surface area. I am interested in the same problem under the constraint that the shape must lie between the planes ...
4
votes
1
answer
115
views
Nearest point is always regular for isoperimetric hypersurfaces
In his paper "Paul Levy's Isoperimetric Inequality" (published as appendix C in Metric Structures for Riemannian and Non-riemannian Spaces), Gromov claims that if $H$ is a minimal $n$-...
4
votes
1
answer
661
views
Generalized widths and reverse Urysohn inequalities
This question is inspired by the discussion in MO questions "Local minimum from directional derivatives in the space of convex bodies" and "Bodies of constant width?" about generalized notions of ...
4
votes
0
answers
140
views
A variation of Zuk's isoperimetric inequality for groups
$\DeclareMathOperator\diam{diam}\DeclareMathOperator\inrad{inrad}$There is a isoperimetric inequality (conjectured by Sikorav and proven by Żuk (Topology 39 (2000) 947–956) which holds in every Cayley ...
4
votes
0
answers
121
views
Integrating a function of distance between a set and its neighbourhood
I am aware of the isoperimetric inequality, which states that if you fix the Lebesgue measure of a measurable set $A \subset \mathbb R^d. d \geq 2$ then the smallest possible value of the perimeter of ...
4
votes
0
answers
126
views
Area lower bound given a mean curvature upper bound?
If $\Sigma$ is a smooth embedded closed hypersurface in $\mathbb R^n$ with (normalized) mean curvature $H\le 1$ (the mean curvature of the unit sphere), then its ($(n-1)$-dimensional) area is at least ...