Rotations around \(n-3\) chords \(d_i\) by \(n-3\) angles \(\theta_i\) commute.

The \((n-3)\)-dimensional  moment polytope \(\mathcal{P}_n \subset \mathbb{R}^{n-3}\) is defined by the triangle inequalities

There exists an almost-everywhere defined map \(\alpha: \mathcal{P}_n \times (S^1)^{n-3} \to \text{Pol}_3(n)/SO(3)\).

Sampling polygons in \(\mathbb{R}^3\)

Theorem (with Cantarella, 2016)

The map \(\alpha\) pushes forward the standard probability measure on \(\mathcal{P}_n \times (S^1)^{n-3}\) to the correct probability measure on \(\text{Pol}_3(n)/SO(3)\).

Corollary

Independently sampling \(\mathcal{P}_n\) and \((S^1)^{n-3}\) is a perfect sampling algorithm for equilateral \(n\)-gons in \(\mathbb{R}^3\).

1. Find a natural map \(g:\text{Arm}_d(n) \to \text{Pol}_d(n)\).
2. Sample points in \(\text{Arm}_d(n)\) and apply \(g\).
3. Hope the pushforward measure is almost uniform.

Idea: \(g\) should map each arm to the closed polygon which is closest to it.

Problem: This is not well-defined on all of \(\text{Arm}_d(n)\).

Definition

A geometric median (or Fermat-Weber point) of a collection \(X=\{x_1,\ldots , x_n\}\) of points in \(\mathbb{R}^d\) is any point closest to the \(x_i\):

\(\text{gm}(X)=\text{argmin}_y \sum \|x_i-y\|\)

Definition

A point cloud has a nice geometric median if:

• \(\text{gm}(X)\) is unique (\(\Leftarrow X\) is not linear)
• \(\text{gm}(X)\) is not one of the \(x_i\)

Definition

If the edge cloud \(X\) of an equilateral arm has a nice geometric median, the geometric median closure \(\text{gmc}(X)\) recenters the edge cloud at the geometric median.

\(\text{gmc}(X)_i = \frac{x_i-\text{gm}(X)}{\|x_i - \text{gm}(X)\|}\)

Proof

\(\text{gm}(X)\) minimizes the total distance function

\(d_X(y) = \sum_i \|x_i - y\|\),

which is convex everywhere and smooth away from the \(x_i\), and

\(\nabla d_X(y) = \sum_i \frac{x_i-y}{\|x_i-y\|}\).

Definition

An arm or polygon \(X\) is given by \(n\) edge vectors \(x_i \in \mathbb{R}^d\), or a single point in \(\mathbb{R}^{dn}\). The distance between \(X\) and \(Y\) is the Euclidean distance between these points in \(\mathbb{R}^{dn}\).

Theorem* (with Cantarella)

The fraction of \(\text{Arm}_d(n)\) without a geometric median closure \(\to 0\) exponentially fast in \(n\).

For a random point cloud \(X = (x_1, \ldots , x_n)\) on \(S^{d-1}\), want to show \(\text{gm}(X)\) is close to the origin (and therefore \(\text{gmc}(X)\) exists) with high probability.

Theorem (Bernstein inequality for Hilbert spaces)

Let \((\Omega,\mathcal{A},P\) be a probability space, \(H\) a separable Hilbert space, \(B > 0, \sigma > 0\). If \(\xi_1,\ldots , \xi_n:\Omega \to H\) are independent r.v.'s satisfying \(\mathbb{E}(\xi_i)=0, \|\xi_i\|_\infty \leq B, \mathbb{E}(\|\xi_i\|_H^2)\leq \sigma^2\), then

\(P\left(\left\|\frac{1}{n}\sum \xi_i\right\|_H \geq \sqrt{\frac{2\sigma^2 \tau}{n}} + \sqrt{\frac{\sigma^2}{n}} + \frac{2B\tau}{3n}\right) \leq e^{-\tau}\)

In our case, let \(\xi_i(p) = \|x_i-p\| - \text{Ad}(p)\) and \(H = L^2(B^d)\). Then, e.g., 

\(P\left(\left\|\frac{1}{n}d_X-\text{ad}\right\|_2 > \frac{30}{n^{1/3}}\right) \leq e^{-n^{1/3}} \)

In other words, \(d_X\) and \(d \text{Ad}\) are close in \(L^2\) with high probability

Proposition (with Cantarella)

If \(f\) is differentiable on \(B^d\) with \(\|\nabla f\|_\infty \leq K\), then

Since \(\|f\|_1 \leq \text{Vol}(B^d) \|f\|_2\), we can combine this with the concentration result to see that \(d_X\) is \(L^\infty\) close to the average distance function with high probability, and hence the Hjort & Pollard lemma applies.

Let \(\phi_n(\ell)\) be the density of the end-to-end distance in an \(n\)-step random flight. From Lord Rayleigh,

This is piecewise-polynomial of degree \(n-3\).

Proposition

The pdf of the length of the chord connecting \(v_1\) to \(v_{k+1}\) in an \(n\)-gon is a constant multiple of

Conjecture

The probability measure generated by geometric median sampling converges (exponentially fast?) to the uniform distribution on equilateral \(n\)-gons as \(n \to \infty\).

Conjecture

The integral of any function which varies slowly enough with respect to any permutation-invariant probability measure on equilateral \(n\)-gons converges to the integral of the function with respect to the uniform distribution on equilateral \(n\)-gons as \(n \to \infty\).

Algorithm

To follow a flow in polygon space:

1. Compute the infinitesimal variation \(V\) in \(\mathbb{R}^{dn}\) at \(X^{(n)}\) tangent to \(\text{Pol}_d(n)\) (or at least \(\text{Arm}_d(n)\)).
2. Follow the exponential map in \(\text{Arm}_d(n)\) in direction \(V\) (to preserve edgelengths exactly).
3. Use geometric median closure to get \(X^{(n+1)} \in \text{Pol}_d(n)\).

Definition

If \(P\) is a closed \(n\)-gon in \(\mathbb{R}^d\) with edge cloud \(X\) and \(\lambda_1\) is the first eigenvalue of \(X^TX\), define

Theorem (with Cantarella)

If \(P\) is a closed equilateral \(n\)-gon, every arm within \(d_{\text{safe}}\) of \(P\) has a geometric median closure. This distance bound is positive if \(P\) is not contained in a line.

Algorithm

To follow a flow in polygon space:

1. Compute the infinitesimal variation \(V\) in \(\mathbb{R}^{dn}\) at \(X^{(n)}\) tangent to \(\text{Pol}_d(n)\) (or at least \(\text{Arm}_d(n)\)).
2. Find \(\lambda_1\) and use it to compute \(d_{\text{safe}}(X^{(n)})\).
3. Follow the exponential map in \(\text{Arm}_d(n)\) in direction \(V\) (to preserve edgelengths exactly), but don't go further than \(d_{\text{safe}}\).
4. Use geometric median closure to get \(X^{(n+1)} \in \text{Pol}_d(n)\).