Fiber Polytopes

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“…Nineteen of these lie in the mixed cells of A w + 6. The four remaining points are (2,5), (2,6), (1,3), (1,4). If we order the set £ such that these four extraneous points come first, then our matrix has the structure where I 4 , denotes the 4 x 4-unit matrix and 0 denotes the 19 x 4-zero matrix.…”

“…We first need to recall some fine print of polytope theory. A polyhedral subdivision (such as a mixed decomposition, or a triangulation) is always a collection of labeled subsets of the given labeled multiset of points [3,12]. Thus each facet F = F 0 + F 1 + • • • + F n of a CMD A u is equipped with additional combinatorial data, consisting in a sequence of subsets (A 0 , A 1 , ..., A n ), where A i C A i and F f = conv(A').…”

“…, y n ) and we form the auxiliary polynomial Its support is the 2n-dimensional set Our first theorem in this section is a purely combinatorial result about secondary polytopes and fiber polytopes (cf. [3,6] £(A, Q) under a 2 (cf. [3, Lemma 2.3]).…”

“…Two polytopes are called strongly isomorphic if they lie in the same affine space and they have the same normal fan. (In [3] we used the term normally equivalent). This implies that they have the same face lattice, but it is stronger.…”

“…., A n . Theorem 5.4 states that the secondary polytope S(A) is strongly isomorphic to a certain fiber polytope, which, in the notation of [3], can be expressed as We show that the resultant polytope N(R) is a Minkowski summand of (2).…”