Minkowski sums of convex lattice polytopes
نویسنده
چکیده
submitted at the Oberwolfach Conference “Combinatorial Convexity and Algebraic Geometry” 26.10–01.11, 1997 Throughout, we fix the notation M := Z and MR := R . Given convex lattice polytopes P, P ′ ⊂ MR, we have (M ∩ P ) + (M ∩ P ) ⊂ M ∩ (P + P ), where P + P ′ is the Minkowski sum of P and P , while the left hand side means {m+m | m ∈ M ∩ P,m ∈ M ∩ P }. Problem 1 For convex lattice polytopes P, P ′ ⊂ MR when do we have the equality (M ∩ P ) + (M ∩ P ) = M ∩ (P + P )? We always have the equality if r = 1. This need not be the case, however, if r ≥ 2 as the following example shows: .................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
منابع مشابه
On f-vectors of Minkowski additions of convex polytopes
The objective of this paper is to present two types of results on Minkowski sums of convex polytopes. The first is about a special class of polytopes we call perfectly centered and the combinatorial properties of the Minkowski sum with their own dual. In particular, we have a characterization of face lattice of the sum in terms of the face lattice of a given perfectly centered polytope. Exact f...
متن کاملf-Vectors of Minkowski Additions of Convex Polytopes
The objective of this paper is to present two types of results on Minkowski sums of convex polytopes. The first is about a special class of polytopes called perfectly centered and the combinatorial properties of the Minkowski sum with their own dual. In particular, we have a characterization of faces of the sum in terms of the face lattice of a given perfectly centered polytope. Exact face coun...
متن کاملMinkowski Sum of Polytopes and Its Normality
In this paper, we consider the normality or the integer decomposition property (IDP, for short) for Minkowski sums of integral convex polytopes. We discuss some properties on the toric rings associated with Minkowski sums of integral convex polytopes. We also study Minkowski sums of edge polytopes and give a sufficient condition for Minkowski sums of edge polytopes to have IDP.
متن کاملMinkowski sum of HV-polytopes in Rn
Minkowski sums cover a wide range of applications in many different fields like algebra, morphing, robotics, mechanical CAD/CAM systems ... This paper deals with sums of polytopes in a n dimensional space provided that both H-representation and V-representation are available i.e. the polytopes are described by both their half-spaces and vertices. The first method uses the polytope normal fans a...
متن کاملLattice polytopes cut out by root systems and the Koszul property
We show that lattice polytopes cut out by root systems of classical type are normal and Koszul, generalizing a well-known result of Bruns, Gubeladze, and Trung in type A. We prove similar results for Cayley sums of collections of polytopes whose Minkowski sums are cut out by root systems. The proofs are based on a combinatorial characterization of diagonally split toric varieties.
متن کامل