Relative Stanley–reisner Theory and Lower Bound Theorems for Minkowski Sums
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چکیده
This note complements an earlier paper of the author by providing a lower bound theorem for Minkowski sums of polytopes. In [AS16], we showed an analogue of McMullen’s Upper Bound theorem for Minkowski sums of polytopes, estimating the maximal complexity of such a sum of polytopes. A common question in reaction to that research was the question for an analogue for the Barnette’s Lower Bound Theorem in the same setting, and the purpose of this note is to put the question to rest. Lower Bound Theorem for polytopes. For a simplicial d-dimensional polytope P on n vertices and 0 ≤ k < d fk(P ) ≥ fk(Stackd(n)) where Stackd(n) is a d-dimensional cyclic polytope on n vertices. Moreover, equality holds for all k whenever it holds for some k0, k0 + 1 ≥ bd2c. This theorem was proven by Barnette [Bar71], and is substantially deeper than the Upper Bound Theorem as it relates to the standard conjectures for toric varieties, see also [Kal87, MN13, Adi17]. In this paper we will address more general lower bound problems for polytopes and polytopal complexes. Recall that the Minkowski sum of polytopes P,Q ⊆ Rd is the polytope P +Q = {p+ q : p ∈ P, q ∈ Q}, and bounding the complexity of Minkowski sums of polytopes is an important problem in several fields of mathematics, see [AS16] for a more thorough discussion of the applications. We thus address the question: For given k < d and n1, n2, . . . , nm, what is the minimal number of k-dimensional faces of the Minkowski sum P1 + P2 + · · ·+ Pm for polytopes P1, . . . , Pm ⊆ Rd with vertex numbers f0(Pi) = ni for i = 1, . . . ,m? Date: November 17, 2017. 2010 Mathematics Subject Classification. 52B05, 13F55, 13H10, 05E45.
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تاریخ انتشار 2017