The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound

In the setting of The Flat Torus: Standing Notation, used with $n=d$ for a natural number $d$ with $1\le d$, and of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $\varpi$ be the wrapped displacement and $d_{\mathbb{T}}$ the flat torus distance. Let $Q$ be the half-open unit cell, $\pi$ the wrapping map and $\mathbb{Z}^{d}$ the integer lattice. ¶Call $z\in\mathbb{R}^{d}$ regular if $z_{i}-\tfrac12\notin\mathbb{Z}$ for every $i\in[d]$. Then the following hold for all $x,y,z\in\mathbb{R}^{d}$ and $k,m\in\mathbb{Z}^{d}$.

1. (Range)¶ $-\tfrac12\le\varpi(z)_{i}<\tfrac12$ for every $i\in[d]$; $z-\varpi(z)\in\mathbb{Z}^{d}$; $\varpi(z+k)=\varpi(z)$; and $\lVert\varpi(z)\rVert^{2}\le d/4$.

2. (Minimality)¶ $\lVert\varpi(z)\rVert\le\lVert z-k\rVert$. Consequently $d_{\mathbb{T}}(x,y)\le\lVert y-x-k\rVert$ for every $k\in\mathbb{Z}^{d}$, with equality for $k=y-x-\varpi(y-x)$; in particular $d_{\mathbb{T}}(x,y)\le\lVert x-y\rVert$.

3. (Regular points)¶ Put $r(z)=\min_{i\in[d]}\bigl(\tfrac12-|\varpi(z)_{i}|\bigr)$, the least of these $d$ real numbers (an iterated minimum of two). Then $r(z)\ge0$, and $z$ is regular if and only if $r(z)>0$. If $z$ is regular, then $\lVert\varpi(z)\rVert<\lVert z-k\rVert$ for every $k\in\mathbb{Z}^{d}$ with $k\ne z-\varpi(z)$; $\varpi(-z)=-\varpi(z)$; and every $z'\in\mathbb{R}^{d}$ with $\lVert z'-z\rVert<r(z)$ is regular and satisfies $\varpi(z')=\varpi(z)+(z'-z)$. The map $r:\mathbb{R}^{d}\to\mathbb{R}$ is Borel.

4. (Symmetry and periodicity)¶ $d_{\mathbb{T}}(x,y)=d_{\mathbb{T}}(y,x)$; $d_{\mathbb{T}}(x+k,y+m)=d_{\mathbb{T}}(x,y)$; $d_{\mathbb{T}}(\pi(x),\pi(y))=d_{\mathbb{T}}(x,y)$; and $d_{\mathbb{T}}(x,y)=0$ if and only if $y-x\in\mathbb{Z}^{d}$.

5. (Triangle inequality)¶ $d_{\mathbb{T}}(x,z)\le d_{\mathbb{T}}(x,y)+d_{\mathbb{T}}(y,z)$.

6. (Metric on the cell)¶ The restriction of $d_{\mathbb{T}}$ to $Q\times Q$ is a metric on $Q$.

7. (Lipschitz bound and measurability)¶ For all $x,y,x',y'\in\mathbb{R}^{d}$,

$$|d_{\mathbb{T}}(x,y)-d_{\mathbb{T}}(x',y')|\le\lVert x-x'\rVert+\lVert y-y'\rVert,\qquad |d_{\mathbb{T}}(x,y)^{2}-d_{\mathbb{T}}(x',y)^{2}|\le\sqrt{d}\,\lVert x-x'\rVert .$$

The map $w\mapsto d_{\mathbb{T}}(\mathrm{pr}_{1}(w),\mathrm{pr}_{2}(w))$ on $\mathbb{R}^{d+d}$ and the map $\varpi:\mathbb{R}^{d}\to\mathbb{R}^{d}$ are Borel.