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The sum of squares of chords has proven to be particularly useful in studying arrangements of lines for redundant linear encodings of data. Suppose $n=\binom{d+1}{2}$. In this case, we may identify $ …

Inspired by this question, I would like to determine the probability that a random knot of 6 unit sticks is a trefoil. This naturally leads to the following question: Is there a way to sample unifor …

This is more of a long comment than an answer: Let's assume $f$ is a $k\times d$ matrix with orthogonal rows, each with norm $\sqrt{d/k}$ (random $f$'s of this form are known to satisfy JL). Consider …

The Euclidean distortion of a metric space $X$, denoted $c_2(X)$, is the infimum of $c$ for which there exists a map $f\colon X\to\ell^2$ such that $$d_X(x,y) \leq \|f(x)-f(y)\|_{\ell^2} \leq c\cdot d …

Without loss of generality, $K$ is a polytope with its vertices in the $B_i$'s. By Farkas' lemma, it's equivalent for the vertices to reside on the same side of a subspace of codimension $1$. With thi …

Without loss of generality, the vectors are in general position. Take $d$ vectors with pairwise obtuse angles and rotate them so that they form the columns of a $d\times d$ upper-triangular matrix wit …

You can pack at most $d+1$ pairwise obtuse vectors in $\mathbb{R}^d$. Several proofs of this fact can be found here.

Fix a positive integer $\ell$. For $x_1,\dotsc,x_n\in S^{d-1}$, Venkov proved that $$ \sum_{i=1}^n\sum_{j=1}^n(x_i\cdot x_j)^{2\ell}\geq\frac{(2\ell-1)!!(d-2)!!}{(d+2\ell-2)!!}\cdot n^2, $$ with equal …

Take iid Gaussian random variables $X_1,\ldots,X_d$ with mean $0$ and variance $1/d$. Normalizing the vector $X=(X_1,\ldots,X_d)$ will produce a random point on the unit sphere, but it's already close …

The wall that separates me from my neighbor acts as a low-pass filter. That means I get to listen to the projection of Justin Bieber onto the subspace of low-frequency audio signals.

Pick some small $\epsilon>0$, take the dilated integer lattice $\epsilon^2\cdot\mathbb{Z}^2$ to be our vertex set, and draw an edge between two vertices if their Euclidean distance is between $\epsilo …

Given a nonempty closed set $C\subseteq\mathbb{R}^n$, let $S_C\subseteq\mathbb{R}^n$ denote the set of points for which the closest point in $C$ is not unique. Suppose $C$ satisfies each of the follow …

This is very similar to the cryo-electron microscopy problem: You want to image a certain macromolecule, and the scale of the macromolecule requires the use of an electron microscope. Unfortunately, s …