Given a convex full-dimensional polytope $P\subset\Bbb R^d$ (convex hull of finitely many points and not contained in any proper affine subspace) and a symmetry thereof (a linear map $\smash{T\in\mathrm{GL}(\Bbb R^d)}$ that fixes $P$ set-wise).


This symmetry induces an automorphism $\phi:\mathcal F(P)\to\mathcal F(P)$ of the face lattice $\mathcal F(P)$.



> 
> **Question:** Suppose that
> 
> 
> 1. $\phi$ is an involution, i.e. $\phi\circ\phi=\mathrm{id}$, and
> 2. $\phi$ fixes only the minimal and the maximal element of $\mathcal F(P)$.
> 
> 
> Is it true that $T$ is necessarily the central inversion $-I$ (in particular, $P$ is centrally symmetric)?
> 
> 
> 


I know that centrally symmetric simplicial/polytopal complexes are defined by the existence of such an automorphism.
So I wondered whether this is geoemtrically justified.



A combinatorial characterization of the central inversion of a polytope?
The answer is *Yes* and it also applies not only to convex polytopes, but also to any other bounded full-dimensional polytopal complex with a well-defined interior (not necessarily spherical or convex). For simplicity, say that $P$ contains the origin in its interior.


Suppose that $T\in\mathrm{GL}(\Bbb R^d)$ induces an automorphism $\phi$ with the given properties. As an involution, its only eigenvalues must be $\pm 1$. But if $v\in\Bbb R^d$ were an eigenvector to eigenvalue $1$, then $\Bbb R v$ would intersect a (proper) face of $P$ (this is where I need that $P$ is bounded, full-dimensional with a well-defined interior and $0\in P$). This face is then necessarily fixed by $T$, in contradiction to the second property of $\phi$.


Thus, all eigenvalues of $T$ are $-1$, and hence $T=-I$.


