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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 allpoints $(L_1,\dots,L_5)\in\mathbb{R}^5$ with the following property: thereexists 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 unitoutward normals to the faces of $P$ and the perimeter of the face with theoutward normal $\boldsymbol{n}_k$ is equal to $L_k$ for all $k=1,\dots,5$. Ourmain result reads that $\mathscr{L}$ is not a locally-analytic set, i.\,e., weprove that, for some point $(L_1,\dots,L_5)\in\mathscr{L}$, it is not possibleto 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 thisresult as an obstacle for finding an existence theorem for a compact convexpolytope with prescribed directions and perimeters of the faces.

Why there is no an existence theorem for a convex polytope with prescribed directions and perimeters of the faces?