Voronoi cell

Remarks

Each condition $\abs{x - p} \le \abs{x - q}$ defines a closed half-space, so $V(p)$ is a closed convex set. Example: for $\Lambda = \Z^2$, the cell is $V(0) = [-\tfrac12, \tfrac12]^2$. Indeed, $\abs{x} \le \abs{x - q}$ for $q = (\pm 1, 0)$ and $q = (0, \pm 1)$ gives $\abs{x_1} \le \tfrac12$ and $\abs{x_2} \le \tfrac12$, and these four conditions imply the others. Non-example: the closed square $[0, 1]^2$ is not the Voronoi cell of $0$ in $\Z^2$, because its point $(1, 1)$ is at distance $0$ from $q = (1, 1)$ and at distance $\sqrt 2$ from $0$. For a lattice $\Lambda$, the cell $V(0)$ is a convex polytope, symmetric about $0$, its translates $V(0) + v$ with $v \in \Lambda$ cover $\R^n$ with disjoint interiors, and its volume equals $\det \Lambda$. The cells of all points of $S$ form the Voronoi diagram of $S$.