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 ...

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.

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(\...

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 (...

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 ...

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 $\{...

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 ...

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_{...

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=...

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 ...

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 ...

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\...

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 ...

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, ...

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}(\...

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 ...

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 ...

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 $\...

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 ...

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 ...

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 ...

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.
...

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)...

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. ...

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}\...

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/...

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 ...

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. ...

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 ...

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 :...

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