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Recent Questions - MathOverflow

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On comparing planar convex regions of equal perimeter and area

Definitions:

The Hausdorff distance between two point sets is the greatest of all the distances from a point in one set to the closest point in the other set. Given two planar convex regions $C_1$ and $C_2$ both with unit perimeter, we define the difference between $C_1$ and $C_2$ as the least value of Hausdorff distance between $C_1$ an...

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Nonconvex geometric counting

Fix $d < n$. Let $V \subseteq \mathbb R^n$ be a $d$-dimensional linear subspace. We say that a set $S \subseteq [n]$ is good if there exists a unit-norm $v \in V$ such that $v_i^2 > 1/s$ for all $i \in S$. Let $N$ be the number of good sets.

If $V$ is spanned by $d$ standard basis vector, straightforward combinatorics proves that $$\log N \lesssim \log {...


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Counting partially-symmetric Boolean functions

Let $f:\{0,1\}^B\to \{0,1\}$ and $1\le S\le B$ be an integer. We say that a function is partially symmetric in $S$ variables if there exists indices $1\le i_1<...<i_S\le B$ such for every permutation $\pi\in \mathbb{S}^S$ let $\pi^{\uparrow}\in \mathbb{S}^B$ be the permutation which acts as the identity on every index $j\not\in \{i_s\}_{s=1}^S$ and $\pi^{\uparrow}(i)=\p...


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Absolute continuity of the Karhunen-Loeve expansion coefficients

The Karhunen-Loeve theorem (see these notes or the wikipedia page, for example) states the followin...


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Spectrally undetermined finite orthogonal groups

For a representation $\rho$ of a group $G$, we denote by $\operatorname{tr}(\rho)$, the multiset $\{\{\operatorname{tr}(g):g\in G\}\}$. Similarly, we denote by $\operatorname{sp}(\rho)$ the multiset $\{\{\operatorname{sp}(g):g\in G\}\}$, where $\operatorname{sp}(M)$ is the spectrum of $M$ (which is itself a multiset).

Let $D_{\operatorname{tr}}$ be the collection of al...


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