12 votes
Accepted

A long-lasting conjecture of Pólya & Szegő

I'm pretty sure that this is still open for $n$-gons (with $n\geq 5$). As far as I know, basically no progress has been made since the original proofs for triangles/quadrilaterals. There have been ...
Otis Chodosh's user avatar
  • 7,037
12 votes
Accepted

Isoperimetric inequality on the plane

No, there is no bound --- yinyang is our friend. This example works if $d\cdot c>2$; otherwise there should be an upper bound.
Anton Petrunin's user avatar
10 votes

Is there a volume-preserving diffeomorphism of the disk with prescribed singular values?

This answers the first (simple) half of the question, asking just about a smooth map. In fact, you've already given an answer to it, in some sense. Apply the map $f: re^{i \theta} \to \sigma_1re^{i(\...
Dmitri Panov's user avatar
  • 28.7k
7 votes
Accepted

An isoperimetric-type inequality inside a cube

This result is known as the relative isoperimetric inequality, see e.g. Functions of Bounded Variation by L. Ambrosio, N. Fusco and D. Pallara (2000), Eq. (3.43). It follows from Poincare inequality (...
Skeeve's user avatar
  • 1,277
6 votes
Accepted

Poincare's argument for maximizing the Coulomb energy

H. Poincaré, Sur une théorème de M. Liapounoff relatif a l’équilibre d’une masse fluide, Comptes Rendus de L’Academie des Sciences 104, 622–625 (1887). As discussed by G.C. Evans, Poincaré assumes ...
Carlo Beenakker's user avatar
5 votes
Accepted

area variation of a closed surface under ${\rm SL}(3)$

In fact, for any surface $\Sigma$, there exists a matrix $A_0 \in SL_3(\mathbb{R})$ so that $A_0\cdot\Sigma$ is a critical point for variations by $sl(3)$. The point is that if we look at the orbit $\{...
Ian Agol's user avatar
  • 66.5k
5 votes
Accepted

Work on "Churning Polygons"

If I understand your question correctly, you're asking about (signed) area-preserving deformations of planar polygonal linkages. One place to start reading about polygonal linkages is this chapter by ...
j.c.'s user avatar
  • 13.5k
5 votes
Accepted

Graph which do not satisfy a pseudo-Poincaré inequality

A counterexample is the subgraph of the $\mathbb{Z}^2$ Cayley graph found by taking squares $S_i$ of side $i$ and arranging them along a (near)diagonal in a chain so that each $S_i$ is adjacent to $S_{...
user48047's user avatar
5 votes
Accepted

Isoperimetric type inequality in $\mathbb{R}^2$

The problem of determining the volume of the tubular neighborhood of a submanifold of Euclidean spaces was studied by Hotelling and Weyl (and then others). See https://www.jstor.org/stable/2371513?seq=...
Willie Wong's user avatar
  • 36.5k
5 votes

Mass transportation proof of the Gaussian isoperimetric inequality?

See e.g. Section 2.1 "Talagrand's transport inequalities and Gaussian dimension-free concentration" of Gozlan's survey. Theorem 2.3 there is Talagrand's result that the standard Gaussian ...
Iosif Pinelis's user avatar
4 votes

Area of a elliptic surface confined by a sphere

I do not think so. Just imagine that you peel a large orange whose surface area is much bigger than that of a unit sphere. Then you "spiral" the peel to make it arbitrarily small and place it inside ...
Piotr Hajlasz's user avatar
4 votes

Is the radial projection map area increasing?

Here is another counterexample. This is in $\mathbb{R}^3$ for simplicity, but the same argument works in any dimension. Let $$ S_\epsilon=\{(x,y,z):\, x^2+y^2\geq 0.01,\ z=\epsilon,\ z^2+y^2+z^2\leq 1\...
Piotr Hajlasz's user avatar
4 votes

Is the radial projection map area increasing?

The answer is negative: the area of $P(S)$ is at most the area of the unit sphere, while the area of $S$ can be made arbitrarily high. An $S$ contained in the unit sphere and star-shaped at $0$ can ...
Benoît Kloeckner's user avatar
4 votes

Graph which do not satisfy a pseudo-Poincaré inequality

Why I do not have a relevant counterexample yet, I decided to write an extended comment on related Poincaré and Sobolev inequalities. The purpose was to place the question in the right context, ...
Piotr Hajlasz's user avatar
4 votes
Accepted

An asymptotic version of the Isoperimetric inequality

I may have missed something but it should follow from Bonnesen's inequality, which states that every domain $\Omega\subset\mathbb{R}^2$ satisfies : $$\mathcal{L}(\partial\Omega)^2-4\pi\mathcal{A}(\...
Thomas Richard's user avatar
4 votes

Convexity of Isoperimetric Domains

Using the definition of a Cartan Hadamard as a complete, simply connected manifold having non-positive curvature, a region in a Cartan Hadamard manifold realizing the optimal isoperimetric ratio need ...
Joel Hass's user avatar
  • 871
4 votes
Accepted

Visual proof of convergence for Steiner's symmetrization

(1) Let $$ C = {\rm conv}\ B\bigcup \{x\}$$ where $B$ is a closed ball of center $o$ and $x$ is not in $B$. Consider arc $A\subset B$, $$ \partial B \bigcap {\rm Int}\ C $$ Note that by any Steiner ...
Hee Kwon Lee's user avatar
  • 1,060
4 votes

When is perimeter continuous under Hausdorff convergence?

comment So, as I understand it, you want to rule out "oscilatory" problems like this. A sequence of $C^\infty$ regions that converge to the unit disk, but their perimeters converge to $\...
Gerald Edgar's user avatar
  • 40.1k
4 votes
Accepted

Lower-bound for $\underset{p \le \gamma_d(A) \le q}{\inf} \gamma(A^\epsilon)$, where $\gamma_d$ is the standard gaussian distribution on $\mathbb R^d$

The answer is $\inf_{p \leq \gamma_{d}(A) \leq q} \gamma(A^{\varepsilon}) = \Phi(\Phi^{-1}(p)+\varepsilon)$ where $\Phi(x) = \int_{-\infty}^{x} \frac{e^{-s^{2}/2}}{\sqrt{2\pi}}ds$. Indeed, all you ...
Paata Ivanishvili's user avatar
4 votes

An Indepth Look at Isoperimetry in the Cayley Graph Generated by All Transpositions

This reply is a bit too long for a comment, but it responds mainly to the comment from the OP who wrote that "it surely must be the case that $\Phi(S_t)$ is typically of order 1". The ...
Yuval Peres's user avatar
3 votes

An inequality inspired by the isoperimetric inequality

I will illustrate the Bellman function approach to prove Wirtinger's inequality which, of course, is simpler than the original problem. The advantage of the approach is that it does not use any ...
Paata Ivanishvili's user avatar
3 votes

Visual proof of convergence for Steiner's symmetrization

Just for illustration, here is an animation of one step of a Steiner symmetrization applied to a rasterized version of a polygon.                    ...
Joseph O'Rourke's user avatar
3 votes
Accepted

A question of Ahlswede and Katona: known lower bounds on $\beta(d,n)$?

(0) Preliminaries: (a) factors: above you should change the factors $\frac{1}{4}$ on the rhs of your equations (4) and (5) to $\frac{1}{2}$ (Kündgens distance is half the distance of Ahlswede/Katona)...
esg's user avatar
  • 3,150
2 votes

$L^{p}$ isoperimetric inequalities on the Hamming cube

We just posted a preprint on arXiv with David Beltran and Jose Madrid, where in Corollary 1.7 it is proved that $$ C(p)=1 \quad \text{for all} \quad p \geq 1.06 $$ partially answering the question. ...
Paata Ivanishvili's user avatar
2 votes

$L^{p}$ isoperimetric inequalities on the Hamming cube

The bound $C(1) \geq \frac{1}{\sqrt{2 \pi}}$ follows from "Bobkov's Inequality" [Bob97], which says that for any $f : \{-1,1\}^n \to [0,1]$ it holds that $\mathcal{U}(\mathbb{E}[f]) \leq \mathbb{E}\...
Ryan O'Donnell's user avatar
2 votes

Shannon's proof of the entropy power inequality

Amir Dembo, Thomas Cover and Joy Thomas talk about (and prove) entropy power inequality towards the end of this paper in two different ways: http://www-isl.stanford.edu/~cover/papers/...
john mangual's user avatar
  • 22.6k
2 votes

Isoperimetric inequality in complex hyperbolic space

My answer might be ten years late, but a great advance toward a proof of this conjecture, considering a lower bound on the Hermitian mean curvature, has been made (at least when the boundary is ...
Didier's user avatar
  • 271
2 votes

Isoperimetric inequality for closed curves in $\mathbb{R}^n$

I found one related isoperimetric inequality due to Schoenberg [1]. We say that a closed curve in $\mathbb{R}^{2n}$ is convex provided it never crosses a hyperplane more than $2n$ times. Theorem. ...
Piotr Hajlasz's user avatar
2 votes
Accepted

Compute lower bound on $\min_{E} \mathcal N(0,\sigma^2 I_n)(E)$ subject to $vol(E \cap H_n(r)) / vol(H_n(r)) \ge p$ where $H_n(r)$ is $n$-hemisphere

Let $\nu:=\mathcal N(0,\sigma^2 I_n)$ and $p\in[0,1]$. Concerning Question 1: For the minimizing $E$, without loss of generality (wlog) we clearly have $E\subseteq H_n(r)$ and $\lambda_n(E;r)=p$. It ...
Iosif Pinelis's user avatar
2 votes

What happens to the Gaussian volume of a Borel set when it is translated?

It turns out that Neyman-Pearson theory helps get a nontrivial inequality. Notations. For a p.s.d matrix $M$ of size $p$, consider the inner product on $\mathbb R^p$ defined by $\langle x,z \rangle_M :...
dohmatob's user avatar
  • 6,694

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