# Tropical Convexity

@inproceedings{Develin2003TropicalC, title={Tropical Convexity}, author={Mike Develin and Bernd Sturmfels}, year={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 products of two simplices. Applications to phylogenetic trees are discussed. 2000 Mathematics Subject Classification: 52A30; 92B10

## 245 Citations

Tropical convexity via cellular resolutions

- Mathematics
- 2005

The tropical convex hull of a finite set of points in tropical projective space has a natural structure of a cellular free resolution. Therefore, methods from computational commutative algebra can be…

TROPICAL HYPERPLANE ARRANGEMENTS AND ORIENTED

- Mathematics
- 2008

We study the combinatorial properties of a tropical hyperplane arrangement. We define tropical oriented matroids, and prove that they share many of the properties of ordinary oriented matroids. We…

Tropical hyperplane arrangements and oriented matroids

- Mathematics
- 2007

We study the combinatorial properties of a tropical hyperplane arrangement. We define tropical oriented matroids, and prove that they share many of the properties of ordinary oriented matroids. We…

Triangulations of polytopes and algebraic geometry

- MathematicsISSAC '04
- 2004

The interaction between polyhedral combinatorics and algebraic geometry is a classical theme and two examples from the 70's are the Bernstein-Kouchnirenko-Khovanski Theorem on the number of roots of a generic sparse system via mixed volumes and the algebraic proofs by Stanley of the Upper Bound Theorem for simplicial spheres and the g-theorem for simplified polytopes.

Tropical Ehrhart theory and tropical volume

- MathematicsResearch in the mathematical sciences
- 2020

A novel intrinsic volume concept in tropical geometry is introduced by developing the foundations of a tropical analog of lattice point counting in polytopes by comparing it to existing measures.

Tropical convexity, halfspace arrangements and optimization

- Mathematics
- 2008

This master thesis investigates discrete geometry in the tropical semiring (R, min, +), setting its main focus on convex polytopes and halfspace arrangements. Specifically, tropical analogs to…

On the frontiers of polynomial computations in tropical geometry

- MathematicsJ. Symb. Comput.
- 2006

Fundamental polytopes of metric trees via parallel connections of matroids

- MathematicsEur. J. Comb.
- 2020

Balanced Triangulations of Lattice Polytopes

- Mathematics
- 2005

Regular triangulations of products of lattice polytopes are constructed with the additional property that the dual graphs of the triangulations are bipartite. Such triangulations are instrumental in…

## References

SHOWING 1-10 OF 37 REFERENCES

First steps in tropical geometry

- Mathematics
- 2003

Tropical algebraic geometry is the geometry of the tropical semiring $(\mathbb{R},\min,+)$. Its objects are polyhedral cell complexes which behave like complex algebraic varieties. We give an…

The Cayley Trick and Triangulations of Products of Simplices

- Mathematics
- 2003

We use the Cayley Trick to study polyhedral subdivisions of the product of two simplices. For arbitrary (fixed) l ≥ 2, we show that the numbers of regular and non-regular triangulations ofl × � k…

An explicit computation of the injective hull of certain finite metric spaces in terms of their associated Buneman complex

- Mathematics
- 2002

An explicit computation of the injective hull of certain finite metric spaces in terms of their associated Buneman complex

Tropical Robinson-Schensted-Knuth correspondence and birational Weyl group actions

- Mathematics
- 2002

By using an elementary matrix approach, based on the technique of discrete Toda equation, we construct subtraction-free rational and piecewise linear transformations associated with various…

Combinatorics of hypergeometric functions associated with positive roots

- Mathematics
- 1997

In this paper we study the hypergeometric system on unipotent matrices. This system gives a holonomic D-module. We find the number of independent solutions of this system at a generic point. This…

TOPCOM: Triangulations of Point Configurations and Oriented Matroids

- Computer Science
- 2002

The core algorithms implemented in TOPCOM are described, and implentation issues are discussed.

On Directional Convexity

- Mathematics
- 2001

Motivated by problems from calculus of variations and partial diier-ential equations, we investigate geometric properties of D-convexity. A function f: R d ! R is called D-convex, where D is a set of…