  We choose some special unit vectors $\boldsymbol{n}_1,\dots,\boldsymbol{n}_5$
in $\mathbb{R}^3$ and denote by $\mathscr{L}\subset\mathbb{R}^5$ the set of all
points $(L_1,\dots,L_5)\in\mathbb{R}^5$ with the following property: there
exists a compact convex polytope $P\subset\mathbb{R}^3$ such that the vectors
$\boldsymbol{n}_1,\dots,\boldsymbol{n}_5$ (and no other vector) are unit
outward normals to the faces of $P$ and the perimeter of the face with the
outward normal $\boldsymbol{n}_k$ is equal to $L_k$ for all $k=1,\dots,5$. Our
main result reads that $\mathscr{L}$ is not a locally-analytic set, i.\,e., we
prove that, for some point $(L_1,\dots,L_5)\in\mathscr{L}$, it is not possible
to find a neighborhood $U\subset\mathbb{R}^5$ and an analytic set
$A\subset\mathbb{R}^5$ such that $\mathscr{L}\cap U=A\cap U$. We interpret this
result as an obstacle for finding an existence theorem for a compact convex
polytope with prescribed directions and perimeters of the faces.
