let a convex polytope $\mathcal{P}$ in $E^n$ be defined as in the tag-description with the additional requirement that their volume be strictly positive.
let further the Voronoi Cells $VC(f)$ of $\mathcal{P}$ be defined as the set of points of $\mathcal{P}$ that attain one of their minimal distances to the boundary of $\mathcal{P}$ in facet $f$.
Question:
is it true that the Voronoi cells $VC(f)$ are in turn convex polytopes with strictly positive volume, resp. what are counter examples?
Yes. Denote $H_1,\ldots,H_m$ hyperplanes of facets $f_1,\ldots,f_m$ of $\mathcal{P}$, denote by $\partial \mathcal{P}$ the union of $f_1,\ldots,f_m$ (the boundary of $\mathcal{P}$). Let $d(x,A)$ denote the distance from point $x$ to set $A$.
Proposition. For $x\in \mathcal{P}$ we have $x\in VC(f_j)$ if and only if $d(x,H_j)\leqslant d(x,H_i)$ for all $i=1,\ldots,m$.
Proof. Denote $r=d(x,\partial \mathcal{P})$. Consider the ball centered in $x$ with radius $r$. It contains a point from $f_j$ if and only if it touches $H_j$, otherwise $H_j$ does not intersect this ball. Therefore $d(x,H_i)\geqslant r$ for all $i$, with equality if and only if $x\in VC(f_i)$.
Therefore $VC(f_j)$ is an intersection of finitely many half-spaces (bounded by $H_i$'s and the bisector planes between $H_j$ and $H_i$). So it is a convex polytope. It has positive volume, since for fixed interior point $p$ of $f_j$ all points of $\mathcal{P}$ sufficiently close to $p$ belong to $VC(f_j)$.
