# A Product Formula for the Normalized Volume of Free Sums of Lattice Polytopes

@article{Chen2019APF, title={A Product Formula for the Normalized Volume of Free Sums of Lattice Polytopes}, author={Tianran Chen and Robert Davis}, journal={Advances in Algebra}, year={2019} }

The free sum is a basic geometric operation among convex polytopes. This note focuses on the relationship between the normalized volume of the free sum and that of the summands. In particular, we show that the normalized volume of the free sum of full dimensional polytopes is precisely the product of the normalized volumes of the summands.

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#### References

SHOWING 1-10 OF 32 REFERENCES

Counting lattice points in free sums of polytopes

- Mathematics, Computer ScienceJ. Comb. Theory, Ser. A
- 2017

It is deduced that given a lattice polytope $P$ containing the origin, the problem of computing the number of lattice points in all rational dilates of the free sum of two latticepolytopes contains the origin is equivalent to the Problem of Computing the numberof lattice Points in all integer dilates in all free sums of the polytopes with itself.

Ehrhart Series and Lattice Triangulations

- Mathematics, Computer ScienceDiscret. Comput. Geom.
- 2008

Abstract
We express the generating function for lattice points in a rational polyhedral cone with a simplicial subdivision in terms of multivariate analogues of the h-polynomials of the subdivision…

Generalized multiplicities of edge ideals

- Mathematics
- 2016

We explore connections between the generalized multiplicities of square-free monomial ideals and the combinatorial structure of the underlying hypergraphs using methods of commutative algebra and…

Decompositions of Rational Convex Polytopes

- Mathematics
- 1980

Publisher Summary This chapter describes decompositions of rational convex polytopes. Let P be a d-dimensional convex polytope (or d-polytope) in Rm whose vertices have rational coordinates. The…

The Reflexive Dimension of a Lattice Polytope

- Mathematics
- 2004

Abstract.The reflexive dimension refldim(P) of a lattice polytope P is the minimal integer d so that P is the face of some d-dimensional reflexive polytope. We show that refldim(P) is finite for…

Unmixing the Mixed Volume Computation

- Mathematics, Computer ScienceDiscret. Comput. Geom.
- 2019

It is demonstrated through problems from real world applications that substantial reduction in computational costs can be achieved via this transformation in situations where the convex hull of the union of the polytopes has less complex geometry than the original poly topes.

Gröbner bases and convex polytopes

- Mathematics
- 1995

Grobner basics The state polytope Variation of term orders Toric ideals Enumeration, sampling and integer programming Primitive partition identities Universal Grobner bases Regular triangulations The…

Classification of Reflexive Polyhedra in Three Dimensions

- Physics, Mathematics
- 1998

We present the last missing details of our algorithm for the classification of reflexive polyhedra in arbitrary dimensions. We also present the results of an application of this algorithm to the case…

Weighted projective spaces and reflexive simplices

- Mathematics
- 2002

Abstract. We derive a classification algorithm for reflexive simplices in arbitrary fixed dimension. It is based on the assignment of a weight Q ? ℕn+1 to an integral n-simplex, the construction, up…

Criteria for strict monotonicity of the mixed volume of convex polytopes

- MathematicsAdvances in Geometry
- 2019

Abstract Let P1, …, Pn and Q1, …, Qn be convex polytopes in ℝn with Pi ⊆ Qi. It is well-known that the mixed volume is monotone: V(P1, …, Pn) ≤ V(Q1, …, Qn). We give two criteria for when this…