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.
- 45k
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(\...
- 28.9k
10
votes
Accepted
An inequality that may be of isoperimetric nature
Since $f$ has zero mean, we have $f=F'$ for a continuous $F$ on the circle. Then for the curve $\gamma(t)=(F(t), g(t)) $ the integral $\int \sqrt{F'^2+g'^2}$ is the length, and the integral $\int fg=\...
- 109k
8
votes
On a 3D Gagliardo-Nirenberg inequality
Yes. For each $R>0$ consider $f_R \in H_0^{1}(B_R)$ as the solution of
$$
\Delta f_R= \Delta f\qquad \text{in $B_R$}
$$
Existence follows from Lax-Milgram (or any other variational method) since
$$
...
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 (...
- 1,277
7
votes
A long-lasting conjecture of Pólya & Szegő
Together with D. Bucur we propose a strategy which could prove the conjecture for a fixed $n \geq 5$ using a finite number of certified numerical computations.
Our paper can be found here: On the ...
- 2,222
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 ...
- 188k
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 $\{...
- 68.9k
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 ...
- 13.6k
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_{...
- 66
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=...
- 39k
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 ...
- 128k
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}(\...
- 4,123
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 ...
- 28k
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
$$
S_\epsilon=\{(x,y,z):\, x^2+y^2\geq 0.01,\ z=\epsilon,\ z^2+y^2+z^2\leq 1\...
- 28k
4
votes
Is the radial projection map area increasing?
The answer is negative: the area of $P(S)$ is at most the area of the unit sphere, while the area of $S$ can be made arbitrarily high.
An $S$ contained in the unit sphere and star-shaped at $0$ can ...
- 14.4k
4
votes
Graph which do not satisfy a pseudo-Poincaré inequality
Why I do not have a relevant counterexample yet, I decided to write an extended comment on related Poincaré and Sobolev inequalities. The purpose was to place the question in the right context, ...
- 28k
4
votes
Convexity of Isoperimetric Domains
Using the definition of a Cartan Hadamard as a complete, simply connected manifold having non-positive curvature, a region in a Cartan Hadamard manifold realizing the optimal isoperimetric ratio need ...
- 881
4
votes
Accepted
Visual proof of convergence for Steiner's symmetrization
(1) Let
$$ 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 ...
- 1,100
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 $\...
- 41.1k
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 ...
- 3,946
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 ...
- 14.2k
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)...
- 3,255
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.
...
- 151k
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 ...
- 3,946
3
votes
Accepted
Nearest point is always regular for isoperimetric hypersurfaces
I think you're inadvertently opening a big can of worms. The question can be answered by a combination of two facts: the absence of branch points in (almost-)minimising hypersurfaces and Allard's ...
- 5,038
3
votes
On a 3D Gagliardo-Nirenberg inequality
This is a special case of embeddings for homogenuous Sobolev spaces and holds if $u \in L^1_{loc}$ with $\nabla u \in L^p$, $1 \leq p<n$ and the usual $p^*$. A proof of this (and much more) is in ...
- 6,035
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. ...
- 3,946
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}\...
- 6,694
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 ...
- 271
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