Identifying a polytope by its fibre polytopes
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Identifying a polytope by its fibre polytopes Peter McMullen1 Received: 24 November 2019 / Accepted: 21 August 2020 © The Author(s) 2020
Abstract It is shown that a full-dimensional polytope P is uniquely determined by its r dimensional fibre polytopes when r ≥ 2. Further, if r ≥ 4 and the r -dimensional fibre polytopes are zonotopes, then P itself must be a zonotope. Keywords Polytope · fibre polytope · normal fan · zonotope Mathematics Subject Classification 52B11 · 52B12
1 Introduction Fibre polytopes were devised by Billera and Sturmfels (1992) (see also Billera and Sturmfels (1994)), generalizing the notion of secondary polytopes introduced by Gel’fand et al. (1990). The purpose of these constructions was to describe how different subdivisions of a projected convex polytope arise from the faces of the original. In McMullen (2004), answering a question of McDonald (2002), we modified the original definition in order to allow mixed fibre polytopes. It is the definition of fibre polytope given there that we adopt here. It is a well-known fact that a d-dimensional polytope P can be recovered from its orthogonal projections on r -dimensional subspaces when r ≥ 1; the basic reason is that the support functional of P is given by the support functionals of these projections. Perhaps closer in spirit to what we do here is the solution of Hammer’s X-ray problem in Gardner and McMullen (1980); see also Gardner (2006) for a wider picture. What we shall show is that something analogous happens with fibre polytopes. More precisely, if P is a full-dimensional polytope in the d-dimensional euclidean space V then, when 2 ≤ r < d is fixed, knowing the fibre polytopes fib(P; L) for each r -dimensional linear subspace L of V determines P uniquely.
B 1
Peter McMullen [email protected] University College London, Gower Street, London WC1E 6BT, England
123
Beitr Algebra Geom
It is known that a fibre polytope of a zonotope is a zonotope; see [McMullen (2003), Theorem 14.1]. We shall also show a kind of converse: if r ≥ 4 and the r -dimensional fibre polytopes of P are zonotopes, then P itself is a zonotope. In outline, the rest of the paper is as follows. In Sect. 2, we define fibre polytopes and discuss some of their basic properties. In Sect. 3, we look at a special case, which gives us some insight into the general problem. In Sect. 4, we show how to recover enough of the normal fan of a polytope from those of its fibre polytopes. In Sect. 5, we apply this result to establish the main claim of the paper. Section 6 treats the special case of zonotopes. We then make some additional comments in Sect. 7.
2 Fibre polytopes The setting until Sect. 7 is a d-dimensional euclidean space V . If Gr = Gr (V ) denotes the Grassmannian of r -dimensional linear subspaces of V and L ∈ Gr , then we have a short exact sequence Φ
Ψ
O −→ L −→ V −→ M −→ O, with Φ the isometric injection of L in V and Ψ the orthogonal projection of V on M = L ⊥ . We shall not distinguish between spaces and their duals, so that the dual exact sequence is Φ∗
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