# Tag Info

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 ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.                    ...
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### 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)...
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### $L^{p}$ 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. ...