3.3 Tropical polytopes

Tropical polytopes are geometric objects in tropical geometry, using the tropical semiring. They're defined by tropical convex hulls or intersections of tropical halfspaces, with unique properties distinct from classical polytopes.

Tropical polytopes have combinatorial types, dimensions, faces, and duality. They can be constructed using vertex-facet descriptions, halfspaces, and Minkowski sums. Their connections to classical polytopes and applications in optimization and phylogenetics make them valuable in various fields.

Definition of tropical polytopes

• Tropical polytopes are geometric objects that arise in tropical geometry, a branch of mathematics that studies geometric structures using the tropical semiring $(\mathbb{R} \cup \{-\infty\}, \max, +)$
• They are analogous to classical polytopes but exhibit unique properties due to the use of the tropical semiring
• Tropical polytopes can be defined in terms of tropical convex hulls or as intersections of tropical halfspaces

Tropical convex hull

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• The tropical convex hull of a set of points $V = \{v_1, \ldots, v_n\}$ in $\mathbb{R}^d$ is defined as: $\operatorname{tconv}(V) = \{\max(v_1 + \lambda_1, \ldots, v_n + \lambda_n) : \lambda_1, \ldots, \lambda_n \in \mathbb{R} \cup \{-\infty\}, \max(\lambda_1, \ldots, \lambda_n) = 0\}$
• It is the set of all points that can be expressed as the tropical linear combination of the points in $V$
• The coefficients $\lambda_1, \ldots, \lambda_n$ in the tropical linear combination are required to have a maximum value of 0

Tropical hyperplanes as building blocks

• Tropical hyperplanes are the building blocks of tropical polytopes, similar to how classical hyperplanes define classical polytopes
• A tropical hyperplane is defined by a tropical linear form $a_1 \odot x_1 \oplus \ldots \oplus a_d \odot x_d \oplus a_{d+1}$, where $a_1, \ldots, a_{d+1} \in \mathbb{R}$ and $\odot, \oplus$ denote the tropical multiplication and addition, respectively
• The tropical hyperplane divides the space into two halfspaces, and the intersection of finitely many tropical halfspaces forms a tropical polytope

Properties of tropical polytopes

• Tropical polytopes exhibit unique properties that distinguish them from classical polytopes
• These properties include combinatorial types, dimension and faces, and duality
• Understanding these properties is essential for studying and working with tropical polytopes in various applications

Combinatorial types

• The combinatorial type of a tropical polytope is determined by its face lattice, which captures the incidence relations between its faces
• Two tropical polytopes are said to have the same combinatorial type if their face lattices are isomorphic
• The combinatorial types of tropical polytopes can be classified using the concept of regular subdivisions of the product of simplices

Dimension and faces

• The dimension of a tropical polytope is the maximum number of linearly independent points in the polytope minus one
• Faces of a tropical polytope are defined as the intersections of the polytope with its supporting tropical hyperplanes
• The faces of a tropical polytope form a partially ordered set (poset) under inclusion, known as the face lattice

Duality in tropical polytopes

• Tropical polytopes exhibit a duality analogous to the polar duality in classical polytopes
• The tropical dual of a tropical polytope $P$ is defined as: $P^* = \{w \in \mathbb{R}^d : w \cdot v \leq 0 \text{ for all } v \in P\}$
• The face lattice of the tropical dual polytope is the opposite poset of the face lattice of the original polytope

Construction methods

• There are several methods for constructing tropical polytopes, each providing a different perspective on their structure and properties
• These methods include tropical vertex-facet descriptions, tropical halfspaces and inequalities, and tropical Minkowski sum
• Understanding these construction methods is crucial for working with tropical polytopes in various applications

Tropical vertex-facet descriptions

• A tropical polytope can be described by its vertices and facets, similar to the vertex-facet description of classical polytopes
• The tropical vertex description is given by the tropical convex hull of a set of points (vertices)
• The tropical facet description is given by the intersection of a set of tropical halfspaces (facets)
• The tropical vertex and facet descriptions are dual to each other

Tropical halfspaces and inequalities

• Tropical halfspaces are defined by tropical linear inequalities of the form: $a_1 \odot x_1 \oplus \ldots \oplus a_d \odot x_d \oplus a_{d+1} \leq b_1 \odot x_1 \oplus \ldots \oplus b_d \odot x_d \oplus b_{d+1}$
• The intersection of finitely many tropical halfspaces forms a tropical polytope
• Tropical halfspaces and inequalities provide a way to construct tropical polytopes using linear constraints

Tropical Minkowski sum

• The tropical Minkowski sum of two tropical polytopes $P$ and $Q$ is defined as: $P \oplus Q = \{p \oplus q : p \in P, q \in Q\}$
• It is the set of all points obtained by taking the tropical sum of a point from $P$ and a point from $Q$
• The tropical Minkowski sum of polytopes corresponds to the classical Minkowski sum under the valuation map

Connections to classical polytopes

• Tropical polytopes have strong connections to classical polytopes, and these connections provide insights into their structure and properties
• The connections include the tropical limit of classical polytopes, combinatorial equivalence, and tropical analogues of classical theorems
• Exploring these connections helps bridge the gap between tropical and classical geometry

Tropical limit of classical polytopes

• Classical polytopes can be "tropicalized" by taking their tropical limit
• The tropical limit of a classical polytope is obtained by applying the valuation map to its vertices and taking the tropical convex hull of the resulting points
• The tropical limit preserves many combinatorial properties of the original polytope, such as its face lattice

Combinatorial equivalence

• Two polytopes (classical or tropical) are said to be combinatorially equivalent if their face lattices are isomorphic
• Many classical polytopes have combinatorially equivalent tropical counterparts
• Combinatorial equivalence allows for the transfer of results and insights between classical and tropical polytope theory

Tropical analogues of classical theorems

• Several fundamental theorems in classical polytope theory have tropical analogues
• For example, the tropical version of the Minkowski-Weyl theorem states that every tropical polytope can be described as the tropical convex hull of its vertices or as the intersection of tropical halfspaces
• Other examples include tropical analogues of Farkas' lemma, Carathéodory's theorem, and the Minkowski-Weyl theorem for cones

Applications and examples

• Tropical polytopes have found applications in various fields, including optimization, phylogenetics, and algebraic geometry
• These applications demonstrate the practical relevance of tropical polytope theory and its potential for solving real-world problems
• Exploring specific examples helps solidify the understanding of tropical polytopes and their properties

Tropical linear programming

• Tropical linear programming is an optimization problem that involves minimizing a tropical linear objective function subject to tropical linear constraints
• The feasible region of a tropical linear program is a tropical polytope
• Tropical linear programming has applications in scheduling, network flow problems, and discrete event systems

Phylogenetic trees and tropical polytopes

• Phylogenetic trees, which represent evolutionary relationships among species, can be studied using tropical geometry
• The space of phylogenetic trees can be represented as a tropical Grassmannian, which is a tropical analogue of the classical Grassmannian
• Tropical polytopes arise as projections of the tropical Grassmannian and provide a way to analyze and visualize phylogenetic data

Tropical convexity in optimization

• Tropical convexity, which is based on the properties of tropical polytopes, has applications in optimization problems
• Tropical convex hulls and tropical halfspaces can be used to formulate and solve optimization problems in a tropical setting
• Examples include tropical support vector machines, tropical principal component analysis, and tropical regression analysis