Let $\mathbb{R}^n$ be a vector space with Euclidean inner product and $\mathbb{Z}^n \subset \mathbb{R}^n$ be a lattice. Let $P$ be a full dimension convex lattice polytope in $\mathbb{R}^n$ containing the origin, such that for every top dimensional face $F$ of $P$, the vertices of $F$ forms a $\mathbb{Z}$-basis of . Such polytopes are called smooth Fano polytope, since the face-fan of $P$ defines a smooth fano toric variety.
We say $P$ is sphere-like (for lack of a better name), if for each $1 \leq k \leq n-1$ and for each $k$-dimensional face $F$ of $P$, the perpendicular foot from $0$ to the affine hull of $F$ lands in $F$. (The perpendicular foot from a point $p$ to an affine subspace $V$ in $\mathbb{R}^n$ is a point in $V$ closest to $p$.)
Example: $P=conv\{(1,0), (0,1), (-1,-1)\}$ is sphere-like, however $P'=conv\{(1,2),(0,1),(-1,-3)\}$ is not sphere-like, the perpendicular foot to the face $\{(1,2),(0,1)\}$ lands outside of the face.
Question: For each smooth fano polytope $P \subset \mathbb{R}^n$, is it possible to find a $\varphi \in SL(n, \mathbb{Z})$, such that $\varphi(P)$ is sphere-like?
