# Tropical Ehrhart theory and tropical volume

@article{Loho2020TropicalET, title={Tropical Ehrhart theory and tropical volume}, author={Georg Loho and Matthias Schymura}, journal={Research in the Mathematical Sciences}, year={2020}, volume={7} }

We introduce a novel intrinsic volume concept in tropical geometry. This is achieved by developing the foundations of a tropical analog of lattice point counting in polytopes. We exhibit the basic properties and compare it to existing measures. Our exposition is complemented by a brief study of arising complexity questions.

#### 3 Citations

An Invitation to Tropical Alexandrov Curvature

- Mathematics
- 2021

We study Alexandrov curvature in the tropical projective torus with respect to the tropical metric. Alexandrov curvature is a generalization of classical Riemannian sectional curvature to more… Expand

Multivariate volume, Ehrhart, and $h^*$-polynomials of polytropes

- Mathematics
- 2020

The univariate Ehrhart and $h^*$-polynomials of lattice polytopes have been widely studied. We describe methods from toric geometry for computing multivariate versions of volume, Ehrhart and… Expand

The tropicalization of the entropic barrier

- Mathematics
- 2020

The entropic barrier, studied by Bubeck and Eldan (Proc. Mach. Learn. Research, 2015), is a self-concordant barrier with asymptotically optimal self-concordance parameter. In this paper, we study the… Expand

#### References

SHOWING 1-10 OF 66 REFERENCES

A tropical isoperimetric inequality

- Mathematics
- 2016

We introduce tropical analogues of the notion of volume of polytopes, leading to a tropical version of the (discrete) classical isoperimetric inequality. The planar case is elementary, but a… Expand

Tropical Convexity

- Mathematics
- 2003

The notions of convexity and convex polytopes are introduced in the setting of tropical geometry. Combinatorial types of tropical polytopes are shown to be in bijection with regular triangulations of… Expand

Tropical Algebraic Geometry

- Mathematics
- 2007

Preface.- 1. Introduction to tropical geometry - Images under the logarithm - Amoebas - Tropical curves.- 2. Patchworking of algebraic varieties - Toric geometry - Viro's patchworking method -… Expand

Affine buildings and tropical convexity.

- Mathematics
- 2007

The notion of convexity in tropical geometry is closely related to notions of convexity in the theory of affine buildings. We explore this relationship from a combinatorial and computational… Expand

Tropical Convex Hull Computations

- Mathematics
- 2008

This is a survey on tropical polytopes from the combinatorial point of view and with a focus on algorithms. Tropical convexity is interesting because it relates a number of combinatorial concepts… Expand

Weighted digraphs and tropical cones

- Mathematics
- 2015

This paper is about the combinatorics of finite point configurations in the tropical projective space or, dually, of arrangements of finitely many tropical hyperplanes. Moreover, arrangements of… Expand

Stiefel tropical linear spaces

- Computer Science, Mathematics
- J. Comb. Theory, Ser. A
- 2015

It is proved that the L ( A ) s obtained in this way are dual to certain matroid subdivisions of polytopes of transversal matroids, and their combinatorics to a canonically associated tropical hyperplane arrangement. Expand

Determinantal identities over commutative semirings

- Mathematics
- 2004

Abstract We present a development of determinantal identities over commutative semirings. This includes a generalization of the Cauchy–Binet and Laplace Theorems, as well as results on compound… Expand

THE REAL FIELD WITH CONVERGENT GENERALIZED POWER SERIES

- Mathematics
- 1998

We construct a model complete and o-minimal expansion of the field of real numbers in which each real function given on [0, 1] by a series ∑ cnxn with 0 ≤ αn → ∞ and ∑ |cn|rαn 1 is definable. This… Expand

Tropical bisectors and Voronoi diagrams.

- Mathematics
- 2019

In this paper we initiate the study of tropical Voronoi diagrams. We start out with investigating bisectors of finitely many points with respect to arbitrary polyhedral norms. For this more general… Expand