Isodiametry, Variance, and Regular Simplices from Particle Interactions

Abstract

Consider a collection of particles interacting through an attractive-repulsive potential given as a difference of power laws and normalized so that its unique minimum occurs at unit separation. For a range of exponents corresponding to mild repulsion and strong attraction, we show that the minimum energy configuration is uniquely attained—apart from translations and rotations—by equidistributing the particles over the vertices of a regular top-dimensional simplex (i.e. an equilateral triangle in two dimensions and regular tetrahedron in three). If the attraction is not assumed to be strong, we show that these configurations are at least local energy minimizers in the relevant $d_\infty$ metric from optimal transportation, as are all of the other uncountably many unbalanced configurations with the same support. We infer the existence of phase transitions. The proof is based in part on a simple isodiametric variance bound which characterizes regular simplices; it shows that among probability measures on ${{\mathbf {R}}}^n$ whose supports have at most unit diameter, the variance around the mean is maximized precisely by those measures which assign mass $1/(n+1)$ to each vertex of a (unit-diameter) regular simplex.

Isodiametry, Variance, and Regular Simplices

Our variational characterization of the unit simplex, Theorem 2.1, was discovered using convex analysis and duality in [29]. However, it turns out to be closely related to a classical result of Jung [24], for which a modern proof can be found in Danzer, Grünbaum and Klee [16].

Theorem A.1

(Jung) Let $K \subseteq {{\mathbf {R}}}^n$ be compact with ${{\,\mathrm{diam}\,}}(K)=1$. Then K is contained in a closed ball of radius $r_n=\sqrt{\frac{n}{2n+2}}$. Moreover, K contains the vertices of a unit n-simplex unless it lies in some smaller ball.

In our companion work we showed that our characterization implies Jung’s theorem [29]. In this appendix we show instead that our characterization follows from Jung’s theorem, so that the two results are in some sense equivalent. We are grateful to an anonymous seminar participant for drawing our attention to Jung’s work, and to Tomasz Tkocz [37] who subsequently observed independently from us that our characterization could be inferred using Jung’s theorem. Let us begin with an elementary geometric result based on Lemma 3.4, which concerns higher dimensional generalizations $\Omega \subseteq {{\mathbf {R}}}^n$ of Reuleaux’s triangle and tetrahedron.

Lemma A.2

(On Reuleaux simplices) If $\Delta \subseteq {{\mathbf {R}}}^n$ is the set of vertices of a unit n-simplex centered at $z \in {{\mathbf {R}}}^n$ and $\Omega {:}{=} \cap _{x \in \Delta } \overline{B_1(x)}$, then $\Delta = \Omega \cap {\partial }B_{r_n}(z)$ where $B_r(x)$ denotes the ball of radius r centered at x.

Proof

Let $\Delta = \{x_0,\ldots ,x_n\} \subseteq {{\mathbf {R}}}^n$ be the vertices of a unit n-simplex centered at $z = \frac{1}{n+1}\sum x_i$. Any vectors $y_0,\ldots ,y_n$ in a Hilbert space H satisfy

$$\begin{aligned} |\sum _{i=0}^n y_i|^2 + \sum _{0 \le i<j \le n} |y_i - y_j|^2 = (n+1) \sum _{i=0}^n |y_i|^2. \end{aligned}$$

Given an arbitrary point $x \in \Omega {:}{=} \cap _{x \in \Delta } \overline{B_1(x)}$, taking $y_i=\frac{1}{n+1}(x-x_i)$ and $H={{\mathbf {R}}}^n$ the identity above yields

$$\begin{aligned} |x - z|^2 + \frac{n}{2n+2} = \frac{1}{n+1} \sum _{i=0}^n |x-x_i|^2. \end{aligned}$$

Estimating the right hand side with Lemma 3.4(a) yields $|x-z|^2 \le r_n^2,$ with equality if and only if $x \in \Delta$. Thus $\Omega \subseteq \overline{B_{r_n}(z)}$ and $\Delta = \Omega \cap {\partial }B_{r_n}(z)$ as desired.

Proof of Theorem 2.1

(using Theorem A.1) The representation (1.10) shows the vertices of a standard n-simplex of diameter $\sqrt{2}$ lies on a unique sphere of radius $r_n \sqrt{2}$; thus the vertices of a unit n-simplex lies on a (unique) sphere of radius $r_n$. Assume $d=1$ without loss of generality hereafter. Any probability measure $\mu ^*$ which assigns mass $1/(n+1)$ to each vertex of a unit n-simplex therefore has the desired variance $r_n^2$. Conversely, let $\mu \in {{\mathcal {P}}}({{\mathbf {R}}}^n)$ have support $K=\mathop \mathrm{spt}\mu$ with ${{\,\mathrm{diam}\,}}[K] \le 1$. Jung’s theorem then asserts K is enclosed by a sphere $S={\partial }B_{r}(z)$ of radius $r \le r_n$ centered at some $z \in {{\mathbf {R}}}^n$, and that $r<r_n$ unless K contains a unit n-simplex. The familiar computation

$$\begin{aligned} {{\,\mathrm{{Var}}\,}}(\mu ) + |{\bar{x}}(\mu ) - z|^2 = \int _K |x-z|^2 d\mu (x) \le r^2 \le r_n^2 \end{aligned}$$

(A.1)

shows ${{\,\mathrm{{Var}}\,}}(\mu ) \le r_n^2$. We conclude equidistribution $\mu ^*$ over the vertices of the unit n-simplex has maximal variance subject to the unit diameter constraint on its support. Also, (A.1) shows ${{\,\mathrm{{Var}}\,}}(\mu )<r_n^2$ unless ${\bar{x}}(\mu )=z$ and $r=r_n$. Thus $\mu$ has smaller variance than $\mu ^*$ unless K contains the vertices of a unit n-simplex $\Delta {:}{=} \{x_0,\ldots , x_n\} \subseteq K$.

We henceforth assume ${{\,\mathrm{{Var}}\,}}(\mu )=r_n^2$, so $\Delta \subseteq K = \mathop \mathrm{spt}\mu$ and ${\bar{x}}(\mu )=z$. From ${{\,\mathrm{{Var}}\,}}(\mu )=r_n^2$ and $\mathop \mathrm{spt}\mu \subseteq \overline{B_{r_n}(z)}$ we conclude the full mass of $\mu$ lies at distance $r_n$ from its barycenter $z={\bar{x}}(\mu )$, i.e. $K\subseteq S= \partial B_{r_n}(z)$. On the other hand, ${{\,\mathrm{diam}\,}}(K) \le 1$ and $\Delta \subseteq K$ implies $K \subseteq \Omega$ where $\Omega {:}{=} \cap _{i=0}^n \overline{B_1(x_i)}$. Lemma A.2 therefore implies $K=\mathop \mathrm{spt}\mu \subseteq S \cap \Omega = \Delta$. Now there is a familiar bijection between the convex hull ${{\,\mathrm{conv}\,}}(\Delta )$ and convex combinations of its vertices, c.f. Remark 2.5 [29]. The only convex combination of the vertices of $\Delta$ having barycenter at z assigns equal weights $1/(n+1)$ to each vertex. From ${\bar{x}}(\mu )=z=\frac{1}{n+1} \sum _{i=0}^n x_i$ we deduce $\mu = \frac{1}{n+1} \sum _{i=0}^n \delta _{x_i}$ as desired. $\quad \square$