Smooth Fano polytopes can not be inductively constructed (Q940729)

A smooth Fano \(d\)-polytope is a lattice polytope \(P\) (that is, \(P\) has vertices in the integer lattice \(\mathbb{Z}^d\)), with the origin in its interior, and its facets simplices whose vertices form bases of \(\mathbb{Z}^d\). Such a \(P\) is pseudo-symmetric if it has a facet \(F\) such that \(-F\) is also a facet. Refuting a conjecture by \textit{H. Sato} [Tohoku Math. J., II. Ser. 52, No. 3, 383--413 (2000; Zbl 1028.14015)], the author describes a smooth Fano \(5\)-polytope \(P\) with \(8\) vertices which is not pseudo-symmetric, such that there is no smooth Fano \(5\)-polytope \(Q \subset P\) with \(7\) vertices, and no smooth Fano \(5\)-polytope \(R \supset P\) with \(9\) vertices. Thus \(P\) is isolated, in that it cannot be constructed from another smooth Fano \(5\)-polytope by adding or removing a vertex.