InvolveSubmission

THE TROPICAL SEMIRING IN HIGHER DIMENSIONS

JOHN NORTON AND SANDRA SPIROFF

Abstract. We discuss the generalization, in higher dimensions, of the tropicalsemiring, whose two binary operations on the set of real numbers are defined to

be the minimum and the sum of a pair, respectively. In particular, our objects

are closed convex sets, and for any pair, we take the (closed) convex hull oftheir union and their Minkowski sum, respectively, as the binary operations.

We consider the semiring in several different cases.

Introduction

The original tropical semiring is 〈R∪{∞},⊕,�, 〉, with the two operations definedby

x⊕ y = min(x, y) and x� y = x+ y.

The fact that this is a semiring comes from the lack of “additive”inverses, as theneutral object is infinity. Inspired by the survey article Tropical Mathematics bySpeyer and Sturmfels [8, p. 165], we generalize the tropical semiring to higherdimensions. In particular, our elements are polyhedra, or more generally, closedconvex sets, in Rn with a fixed recession cone, i.e., the directions in which the setrecedes, and the operations defined by taking the (closed) convex hull of the unionand the Minkowski sum. Indeed, when n = 1 and the recession cone is R≥0, thenthis definition reduces to the original tropical semiring: the real numbers x andy represent the sets of solutions to the inequalities t ≥ x and t ≥ y, respectively;i.e., they correspond to the polyhedra in R given by the positive rays with verticesat x, y. In particular, for each, the recession cone is the positive ray emanatingfrom the origin, or R≥0. Clearly, the union of these two sets is represented by theinequality t ≥ min(x, y) and likewise, the Minkowski sum is given by the inequalityt ≥ x+ y. The neutral object is infinity, hence, there are no inverses under ⊕.

As suggested in [8], the set of convex polyhedra in Rn with fixed recession conewill form a semiring. We detail this idea in our paper, considering various recessioncones. In particular, we first consider the case of bounded polyhedra, i.e., convexpolytopes, in Rn. In this case, the recession cone is {0} and the axioms follow quitenicely. Furthermore, we can generalize this case to that of compact (convex) sets inRn. These proofs are the content of the second section1. Prior to that, we providethe necessary background on recession cones and asymptotic cones, and includeexamples to demonstrate the possible pathology of ⊕ and � if the recession cone

1991 Mathematics Subject Classification. 16Y60; 52B11; 52A07.

S. Spiroff is supported by a grant (#245926) from the Simons Foundation.1Section two and part of section three is the basis for J. Norton’s undergraduate thesis for the

Honors College at the University of Mississippi.

1

2 JOHN NORTON AND SANDRA SPIROFF

is not fixed. The main portion of the paper is dedicated to establishing the sevenaxioms of the various semirings, and most especially, those dealing with the closureof the operations. The final section of the paper considers unbounded convex sets.We demonstrate the semirings of convex polyhedra and general convex sets, bothwith recession cone equal to the positive orthant2, where for the latter it is necessaryto take the closed convex hull.

1. Background-Polyhedra, Recession Cones, and Asymptotic Cones

Some general references for the material in this section are the books by T. Rock-afellar [6]; G. Ziegler [10]; and K. C. Border [2] (or [3]).

Definition 1.1. [6, p. 10] A subset P of Rn is convex if it satisfies the followingproperty: for every x, y ∈ P and λ ∈ R, 0 < λ < 1, the element λx+ (1− λ)y ∈ P .

Fact 1.2. [6, Theorem 2.3] For any subset S ⊆ Rn, the convex hull of S, denotedconvS, is the set of all convex combinations of the elements of S; i.e.,

convS = {λ1s1 + · · ·+ λksk : si ∈ S;λi ≥ 0;λ1 + · · ·+ λk = 1; k ∈ N}.Definition 1.3. [6, p. 61] Given a non-empty convex set P in Rn, the recessioncone is the set of all y ∈ Rn such that p + y ∈ P for all p ∈ P . Denoted by 0+P ,the recession cone is the set of all directions in which P recedes; i.e., is unbounded.

Fact 1.4. [6, Theorem 8.4] A non-empty closed convex set P in Rn is bounded ifand only if its recession cone 0+P consists of the zero vector alone.

Example 1.5. In the case of n = 2, the following sets have recession cone equalto the first quadrant Q2

1 = {(x, y) : x, y ≥ 0} of the plane.

(1) P = {(x, y) : x ≥ −5, y ≥ −18, y ≥ − 53x+ 2};

(2) Q = {(x, y) : x ≥ −3, y ≥ −15, y ≥ −6x− 16, y ≥ − 12x− 8};

(3) [6, Example p. 62] {(x, y) : x > 0, y ≥ 1/x}.Definition 1.6. [6, p. 170] [10, p. 28] [1, p. 232] A polyhedral convex set in Rn isone that can be expressed as the intersection of some finite collection of closed halfspaces; i.e., it is the set of solutions to some finite system of inequalities Ax ≤ b.A convex polytope is a bounded polyhedron; i.e., the convex hull of a finite set.

Fact 1.7. [10, Proposition 1.12] If P is a polyhedral convex set in Rn, then 0+Pis the set of solutions to the system of inequalities Ax ≤ 0.

Definition 1.8. [6, p. 162] A point x in a convex set P is an extreme point ifthe only way to express x as the convex combination (1−λ)y+λz for y, z ∈ P and0 < λ < 1 is by taking y = z = x. Denote the set of extreme points of P by ext(P ).

Fact 1.9. [6, Corollary 19.1.1] If P is a polyhedral convex set, then ext(P ) is finite.

In the example above, the first two sets are polyhedra (see Figure 2 on p. 8), butthe third one is not. The finite system of inequalities associated to P is−1 0

0 −1− 5

3 −1

[xy

]≤

518−2

,2The n-dimensional analogue to the first quadrant in the plane.

THE TROPICAL SEMIRING IN HIGHER DIMENSIONS 3

so that 0+P = {(x, y) : x ≥ 0, y ≥ 0}, and ext(P ) = {(−5, 31/3), (12,−18)}.

For a set that is not convex, there is a generalization of the notion of a recessioncone. While we only consider convex sets, this new cone is relevant since the twodefinitions coincide when the convex set is closed, hence we may apply relatedresults in the literature in our cases. We apply this material in the last section.

Definition 1.10. [2, Definition 2.34] The asymptotic cone of a set P in Rn,denoted AP , is the set of all possible limits of sequences of the form {αixi}i, whereeach xi ∈ P , αi > 0, and αi → 0.

Some properties of the asymptotic cone will be necessary to our proof:

Fact 1.11. (See [5, §1.9] or [3, Lemma 4].) The following hold for sets E,F in Rn:

(1) AE is a cone; (6) AE is convex if E is convex;(2) AE ⊆ AF if E ⊆ F ; (7) 0+E = AE if E is closed and convex;(3) 0+E ⊆ AE; (8) AE + AF ⊆ A(E + F ) if E + F is convex;(4) AE ⊆ A(E + F ); (9) a set E is bounded if and only if AE = {0}.(5) AE is closed;

Lemma 1.12. For a set P ⊆ Rn, AP = AP .

Proof. First of all, since P ⊆ P , it follows from Fact 1.11(2) that AP ⊆ AP . Forthe converse, let z ∈ AP , and let ε > 0. Then z = limi→∞ αizi, where zi ∈ P andαi > 0 such that αi → 0. Let N1 ∈ N such that if i ≥ N1, then ρ(z, αizi) < ε/2,where ρ(x, y) denotes the distance3 between x and y. Next, as the αi → 0, there issome N2 ∈ N such that if i ≥ N2, then αi < 1/2. Furthermore, since zi ∈ P , thereis some pi ∈ P such that ρ(zi, pi) < ε. Then for all i ≥ max{N1, N2},

ρ(z, αipi) ≤ ρ(z, αizi) + ρ(αizi, αipi) <ε

2+

12· ε = ε.

Therefore, z = limn→∞ αipi, hence z ∈ AP . �

Example 1.13. [9] In R2, let P = {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1} ∪ {(x, y) : 0 ≤x < 1, y ≥ 1}. Although P is unbounded, 0+P = {0}; however, P is not closed (seeFact 1.7). On the other hand, 0+P = {(0, y) : y ≥ 0} = AP = AP .

As the above definitions and results are important to establishing the closure ofthe operation ⊕, the following definition and result are helpful in establishing theclosure of the operation �.

Definition 1.14. [5, 1.9 m., p. 22] The cones C1, C2, . . . , Ck in Rn are positivelysemi-independent if, for any ci ∈ Ci, the condition c1 + c2 + · · ·+ ck = 0 impliesthat each ci = 0.

Fact 1.15. [3, Theorem 8] For closed and convex sets E,F ⊆ Rn whose asymptoticcones AE and AF are positively semi-independent, the Minkowski sum E + F isclosed and A(E + F ) ⊆ AE + AF .

3This is the Euclidean metric on Rn: ρ(x, y) = ||x− y|| = [(x1 − y1)2 + · · ·+ (xn − yn)2]12 .


Example 1.16. [3, Example 2] In R2, set E = {(x, y) : x > 0, y ≥ 1/x} andF = {(x, y) : x < 0, y ≥ −1/x}. Note that both E and F are closed sets, butE + F = {(x, y) : y > 0}, which is not closed.

Finally, Caratheodory’s Theorem (see, e.g., [7, Theorem 1.1.4]) will be helpful whenconsidering the elements of convex sets.

Fact 1.17. [Caratheodory’s Theorem] If a point x lies in the convex (hull of a) setP ⊆ Rn, then x can be written as a convex combination of no more than n+1 pointsin P ; i.e., there are p0, p1, . . . , pn ∈ P and λi ∈ R≥0 such that λ0 +λ1 + · · ·+λn = 1and x = λ0p0 + · · ·+ λnpn.

2. The Tropical Semiring in Higher Dimensions-Bounded Case

2.1. The Semiring of convex polytopes. Recall that a convex polytope in Rn

is a bounded polyhedral set; i.e., the convex hull of a finite number of points inRn. In particular, these sets are those convex polyhedra in Rn with recession coneequal to the origin.

Example 2.1. A convex polytope and a non-convex set, respectively, in R2.

Figure 1. A(−1, 5), B(4, 2), C(2,−2), D(5,−6), E(−2,−7), F (−8,−1), G(−2, 3)

Theorem 2.2. The set of all convex polytopes P,Q in Rn, with operations shownbelow, is a semiring.

(2.1) P ⊕Q = conv(P ∪Q) P �Q = P +Q = {p+ q, p ∈ P, q ∈ Q}.

Proof. Let P,Q,R be convex polytopes in Rn. In particular, set P = conv(p1, . . . , ps)and Q = conv(q1, . . . , qt).

Axiom 1. The operation ⊕ is closed; i.e., conv(P ∪Q) is a convex polytope4.

We claim that conv(P ∪ Q) = conv(p1, . . . , ps, q1, . . . , qt). Let z ∈ conv(P ∪ Q).By Caratheodory’s Theorem, z =

∑ni=0 λiyi, where each λi ≥ 0,

∑ni=0 λi = 1,

and yi ∈ P ∪ Q. For each yi ∈ P , one can write yi =∑s

j=1 δijpj , where δij ≥ 0

for all j and∑s

j=1 δij = 1. If all yi ∈ P , then z =∑n

i=0

(λi

∑sj=1 δijpj

)=∑s

j=1 (∑n

i=0 λiδij) pj ∈ conv(p1, . . . , ps) ⊆ conv(p1, . . . , ps, q1, . . . , qt); similarly, if

4This fact appears in several books without proof. Therefore, we provide an argument. Foralgorithms that compute the convex hull of a finite set of points in the plane, for example, Graham’s

scan and Jarvis’s march, see, e.g., [4, Chapter 33 Section 3].


all yi ∈ Q. Thus, let m ∈ N, m < n, such that y0, . . . , ym−1 ∈ P \ Q andym, . . . , yn ∈ Q. Then

z =n∑

i=0

λiyi =m−1∑i=0

s∑j=1

λiδijpj

+n∑

i=m

(t∑

k=1

λiδikqk

)

is a convex combination of {p1, . . . , , qt}, hence, conv(P ∪ Q) ⊆ conv(p1, . . . , qt).Since the containment ⊇ is clear, conv(P ∪ Q) = conv(p1, . . . , ps, q1, . . . , qt), andthe latter, by Definition 1.6, is a polytope.

Axiom 2. The operation ⊕ is associative.

Regarding (P ⊕Q)⊕R = P ⊕ (Q⊕R), we wish to prove that

conv[conv(P ∪Q) ∪R] = conv[P ∪ conv(Q ∪R)].

It suffices to show that each of these sets is equal to conv(P ∪Q∪R). Consider theset on the left. Since P ∪ Q ∪ R ⊆ conv(P ∪ Q) ∪ R, we have conv(P ∪ Q ∪ R) ⊆conv[conv(P ∪Q) ∪R].

Conversely, as conv(P ∪Q), R ⊆ conv(P ∪Q∪R), it follows that conv(P ∪Q)∪R ⊆conv(P ∪ Q ∪ R). Take the convex hull of both sides: conv[conv(P ∪ Q) ∪ R] ⊆conv(P ∪Q∪R). This establishes that conv(P ∪Q∪R) = conv[conv(P ∪Q)∪R].The argument for conv[P∪conv(Q∪R)] is analogous, hence conv[conv(P∪Q)∪R] =conv[P ∪ conv(Q ∪R)].

Axiom 3. The operation ⊕ is commutative: this is obvious since we are takingunions of sets and order does not matter.

Axiom 4. There exists a neutral object O such that for any convex polytope P inRn, P ⊕O = O ⊕ P = P .

Define O to be the emptyset ∅. First, note that P ∪ ∅ = ∅ ∪ P = P , hence,P ⊕O = conv(P ∪O) = P . Next, ∅ satisfies the convexity property vacuously, andit can be considered a convex polytope by considering ∅ as the solution set of anyinconsistent system.

Axiom 5. The operation � is closed; i.e., P +Q is a convex polytope.

We claim that P + Q = conv({pj + qk|1 ≤ j ≤ s, 1 ≤ k ≤ t}), as per the hintin [1, proof of Lemma 5.124]. Let p ∈ P and q ∈ Q. Write p =

∑sj=1 λjpj

and q =∑t

k=1 µkqk, where λj , µk ≥ 0 and∑s

j=1 λj = 1 =∑t

k=1 µk. Then,p+ q =

∑sj=1 λjpj +

∑tk=1 µkqk = (

∑tk=1 µk)

∑sj=1 λjpj + (

∑sj=1 λj)

∑tk=1 µkqk =∑s

j=1

∑tk=1 λjµk(pj +qk) is a convex combination of {pj +qk|1 ≤ j ≤ s, 1 ≤ k ≤ t}.

Conversely, let∑n

i=0 λi(xi + yi) be a convex combination of {pj + qk|1 ≤ j ≤s, 1 ≤ k ≤ t}; i.e., xi = pj for some j and yi = qk for some k, λi ≥ 0, and∑n

i=0 λi = 1. Then∑n

i=0 λi(xi + yi) =∑n

i=0 λixi +∑n

i=0 λiyi, where the first sumis in conv(p1, . . . , ps) and the second sum is in conv(q1, . . . , qt). Thus, P + Q =conv({pj +qk|1 ≤ j ≤ s, 1 ≤ k ≤ t}), and the latter, by Definition 1.6, is a polytope.

Axiom 6. The operation � is associative: this is obvious since addition of elementsin Rn is associative.

Axiom 7. The operation � distributes over ⊕; i.e., P�(Q⊕R) = (P�Q)⊕(P�R).


We wish to establish that P + conv(Q ∪R) = conv[(P +Q) ∪ (P +R)].

First of all, take p+ z, where p ∈ P, z ∈ conv(Q ∪R). Then z =∑n

i=0 λiyi, whereλi ≥ 0,

∑ni=0 λi = 1, and yi ∈ Q ∪R. Therefore, we have

p+ z = 1p+n∑

i=0

λiyi =

(n∑

i=0

λi

)p+

n∑i=0

λiyi =n∑

i=0

λi(p+ yi).

The elements p+ yj are in P +Q or P +R, and possibly both. Therefore, the lastexpression is in conv[(P + Q) ∪ (P + R)]; i.e., p + z ∈ conv[(P + Q) ∪ (P + R)].Since p and z are arbitrary, we have shown that

P + conv(Q ∪R) ⊆ conv[(P +Q) ∪ (P +R)].

Conversely, since P +Q, P +R ⊆ P + conv(Q ∪R), it follows that

(P +Q) ∪ (P +R) ⊆ P + conv(Q ∪R).

Take the convex hull of both sides:

conv[(P +Q)∪ (P +R)] ⊆ conv[P + conv(Q∪R)] = conv(P ) + conv[conv(Q∪R)],

where the equality follows by [7, Theorem 1.1.2]. Now since both terms in the lastsum are convex, the expression simply equals P + conv(Q∪R). Therefore, we haveestablished the other inclusion, hence P+conv(Q∪R) = conv[(P+Q)∪(P+R)]. �

2.2. The Semiring of Convex Compact Sets. In this section, we generalizethe above work with convex polytopes to general convex compact subsets of Rn.Of import is the Heine-Borel Theorem (see e.g., [1, Theorem 3.19]):

Fact 2.3. [Heine-Borel Theorem] Subsets of Rn are compact if and only if they areclosed and bounded.

Proposition 2.4. The set of all compact convex sets P,Q in Rn, with the opera-tions as in (2.1), is a semiring.

Proof. We note that the arguments for many of the axioms above do not change.In particular, the neutral object is the empty set. However, the two closure axiomsmust be considered, therefore, let P,Q be compact convex sets in Rn.

Axiom 1. The operation ⊕ is closed; i.e., conv(P ∪Q) is a compact convex set.

The union of finitely many compact sets is compact. Thus, P ∪Q is compact. Next,the convex hull of a compact set in Rn remains compact (see, e.g., [1, Corollary5.18]), thus, conv(P ∪Q) is a compact convex set.

Axiom 5. The operation � is closed; i.e., P +Q is a compact convex set.

The Minkowski sum of two convex sets is convex (see, e.g., [7, Theorem 1.1.2]). Asper [3, Corollary 11], the summation of a closed set and a compact set is closed. Assuch, P �Q = P + Q is closed. The fact that P + Q is bounded follows from thefact that P,Q are bounded5. In particular, if ||p|| ≤M1 for all p ∈ P and ||q|| ≤M2

for all q ∈ Q, then ||p + q|| ≤ ||p|| + ||q|| ≤ M1 + M2. Thus, by the Heine-BorelTheorem, P +Q is compact, convex in Rn. �

5This is the Euclidean norm on Rn: ||x|| = [x21 + · · ·+ x2

n]12 .


3. The Tropical Semiring in Higher Dimensions–Unbounded Case

3.1. The Semiring of Convex Polyhedra. We consider the set of convex poly-hedra in Rn with the operations ⊕ and � as in (2.1). Although convex polyhedraare necessarily closed, by definition, the convex hull of two convex polyhedral setsneed not be polyhedral or closed, as evinced by Example 3.1 below; that is, if theirrecession cones do not coincide. Therefore, we restrict our sets to those with thesame recession cone; in particular, we take the positive orthant Qn

1 (the n-th di-mensional equivalent to the first quadrant in the plane), which is a generalizationof the positive ray in the original tropical semiring.

Example 3.1. See [6, p. 177] In R2, let P = {(−1, 0)} and Q = {(x, y) : x, y ≥ 0}.Then conv(P∪Q) = {(−1, 0)}∪{(x, y) : −1 < x, 0 ≤ y}, which is neither polyhedralnor closed. However, 0+P is the origin, while 0+Q = Q2

1.

Theorem 3.2. The set of all convex polyhedra in Rn ∪ {∞} with recession coneequal to the positive orthant Qn

1 and operations defined as in (2.1) is a semiring.

Proof. Let P,Q,R be convex polyhedra in Rn ∪ {∞} with recession cone equal toQn

1 . It suffices to address the axioms regarding closure and the neutral object, sincethe arguments for the remaining axioms apply here.

Axiom 1. The operation ⊕ is closed; i.e., conv(P ∪Q) is a convex polyhedron withrecession cone equal to Qn

1 .

Since conv(P ∪Q) is convex, it remains to establish that conv(P ∪Q) is polyhedralwith a recession cone equal to the positive orthant. Recall that, by [10, Theorem1.2], if ext(P ) is the set of (necessarily finite) extreme points of P , then P =conv(ext(P )) + 0+P , where 0+P = Qn

1 ; likewise Q = conv(ext(Q)) + Qn1 .

First of all, we consider P∪Q, which is not necessarily convex or polyhedral, and thedirections in which it recedes. By Definition 1.6, P is the irredundant intersectionof some finite collection of closed half spaces. By [6, Theorem 8.3], if the half-line{p + λy|λ ≥ 0} is contained in P for any particular p ∈ P and y ∈ Rn, then thesame holds for every other point of P . Since 0+P = Qn

1 , the only directions inwhich P recedes are {(x1, . . . , xn) : xi ≥ 0 for all i}. In particular, P recedes inthe direction of each ei = (0, . . . , 0, 1, 0, . . . , 0), the i-th standard basis vector. Thismeans that the half-spaces of the form

{x|〈x, (0, . . . , 0, 1, 0, . . . , 0)〉 ≥ ai} for some ai ∈ R,

are included in the set of irredundant half-spaces defining P ; i.e., xi ≥ ai. Likewise,0+Q = Qn

1 , hence, for each i, one of the half-spaces defining Q is xi ≥ ci for someci ∈ R. Thus, every element of P ∪Q satisfies the following set of inequalities:

(3.1) {x|〈x, (0, . . . , 0, 1, 0, . . . , 0)〉 ≥ min(ai, ci)}.

Moreover, if z ∈ conv(P ∪ Q) \ (P ∪ Q), then z is in the finite region boundedby the extreme points of P and Q. (See, e.g., the special case of n = 2 below.)Thus, conv(P ∪ Q) ⊆ P ∪ Q ∪ conv(ext(P ) ∪ ext(Q)). The opposite inclusion isobvious. Therefore, noting that 0+(conv(P ∪Q)) = Qn

1 , as per (3.1), we may writeconv(P ∪Q) = conv(ext(P ) ∪ ext(Q)) + Qn

1 , and the latter, by [10, Theorem 1.2],is polyhedral.


The specific case of n = 2.

As above, every element of P ∪Q satisfies the following two inequalities:

{(x, y)|〈(x, y), (1, 0)〉 ≥ min(a1, c1)} {(x, y)|〈(x, y), (0, 1)〉 ≥ min(a2, c2)},

and since P and Q are each defined by intersections of finitely many half spaces,there are at most finitely many additional inequalities. These take the form ofmix− y ≤ bi, where mi < 0, as the recession cones are Q2

1. See the example below:

Figure 2. The graphs of polyhedra P and Q from Example 1.5.

Figure 3. The graphs of P ∪Q and conv(P ∪Q).

It is easy to see that conv(P ∪Q) = conv(ext(P )∪ ext(Q))+ Q21, and in particular,

that conv(P ∪Q) is polyhedral.

Axiom 4. There exists a neutral object O ⊆ Rn∪{∞} such that for all polyhedralconvex sets P with recession cone equal to Qn

1 , P ⊕O = O ⊕ P = P .

As in the case of n = 1 for the original tropical semiring, the neutral object isinfinity, since P recedes in all directions of the positive orthant.

Axiom 5. The operation � is closed; i.e., P + Q is a convex polyhedron withrecession cone equal to Qn

1 .

By [6, Corollary 19.3.2], the Minkowski sum of two polyhedral convex sets in Rn ispolyhedral, and it is convex (see, e.g., [7, Theorem 1.1.2]). Therefore, it remains toshow that 0+(P +Q) = Qn

1 . Since polyhedral convex sets are closed, their recessioncones are equal to their asymptotic cones. Hence by Fact 1.11(8), AP + AQ ⊆A(P + Q). On the other hand, clearly AP ∩ (−AQ) = {0}, thus, by Fact 1.15,A(P+Q) ⊆ AP+AQ; i.e., A(P+Q) = AP+AQ = Qn

1 , and the result follows. �


3.2. The Semiring of Closed Convex Sets. In this section, we generalize theabove work with convex polyhedra to general closed convex subsets of Rn withrecession cone equal to the positive orthant. As evinced in Example 1.13, pathologyarises if the sets are not assumed to be closed; however, despite taking two convexsets that are closed, neither the convex hull of the union nor the Minkowski sumneed be closed, as per Examples 3.1 and 1.16, respectively. Therefore, in this sectionwe take our definition of ⊕ to be the closed convex hull of the union and make use ofthe material at the end of Section 1 in order to ensure the closure of the Minkowskisum. In particular, we have:

Theorem 3.3. The set of all closed convex sets in Rn ∪ {∞} with recession coneequal to the positive orthant and operations defined below is a semiring.

P ⊕Q = conv(P ∪Q) and P �Q = {p+ q|p ∈ P, q ∈ Q},

where conv denotes the closed convex hull of P and Q.

Remark 3.4. Note that conv(P ∪Q) = conv(P ∪Q), as per [1, p. 171].

Proof. Let P,Q,R be closed convex sets of Rn with recession cone equal to thepositive orthant, denoted Qn

1 .

Axiom 1. The operation ⊕ is closed; i.e., conv(P ∪ Q) is a closed convex subsetwith recession cone equal to Qn

1 .

First of all, as ⊕ is defined as the closed convex hull of the union, it only remainsto show that 0+(conv(P ∪Q)) = Qn

1 . From Facts 1.11 and 1.12, it follows that

0+(conv(P ∪Q)) ⊆ A(conv(P ∪Q)) = A(conv(P ∪Q)) = 0+(conv(P ∪Q)),

therefore, it suffices to show Qn1 ⊆ 0+(conv(P ∪Q)) and A(conv(P ∪Q)) ⊆ Qn

1 .

For the right inclusion, take w ∈ Qn1 , and let z ∈ conv(P ∪Q). By Caratheodory’s

Theorem, z can be expressed as a convex combination of exactly n + 1 elementsfrom P ∪Q. In particular, write

z =n∑

i=0

λiyi, where λi ≥ 0,n+1∑i=0

λi = 1, yi ∈ P ∪Q.

Note that

w + z = 1w +n∑

i=0

λiyi =

(n∑

i=0

λi

)w +

n∑i=0

λiyi =n∑

i=0

λi(w + yi).

If yi ∈ P , then w + yi ∈ P , and likewise, if yi ∈ Q, then w + yi ∈ Q. In anycase, w + yi ∈ P ∪ Q for all i. Consequently, by definition of the convex hull,

n∑i=0

λi(w + yi) ∈ conv(P ∪Q); i.e., w + z ∈ conv(P ∪Q), as desired.

Conversely, let z ∈ A(conv(P ∪ Q)). By definition, z = limi→∞ αizi where zi ∈conv(P ∪Q) and αi > 0 such that αi → 0. As above, each zi can be expressed asa convex combination of exactly n+ 1 elements from P ∪Q. In particular,

(3.2) zi =n∑

j=0

λijzij , where λij ≥ 0,n∑

j=0

λij = 1, and zij ∈ P ∪Q.


For a fixed j, consider the sequence {αi(λijzij)}i. Since the αi are non-negativereal numbers converging to 0 and 0 ≤ λij ≤ 1, the coefficients αiλij also convergeto 0 from above. Thus, as infinitely many zij are either in P or in Q, it followsthat limi→∞(αiλij)zij , if it exists, is an element of Qn

1 since AP = 0+P andAQ = 0+Q, by Fact 1.11(7). Assume, for the moment, that all such limits exist,and set rj = limi→∞(αiλij)zij . Then

z = limi→∞

αizi = limi→∞

αi

n∑j=0

λijzij

=n∑

j=0

limi→∞

αiλijzij =n∑

j=0

rj ∈ Qn1 ,

and A(conv(P ∪Q)) ⊆ Qn1 , as desired.

However, as it can not simply be assumed that all of the above limits exist, at thispoint we take a different tack. Write z = (x01, . . . , x0n), zi = (xi1, . . . , xin), andzij = (xij1, . . . , xijn), so that equation (3.2) can be rewritten as

(3.3) (x01, . . . , x0n) = limi→∞

αi(xi1, . . . , xin) = limi→∞

αi

n∑j=0

λij(xij1, . . . , xijn)

.

Suppose that x01 < 0, as the general case is analogous. Then αixi01 < 0 for all i >>0. Since the operation on Rn is performed componentwise, x01 = limi→∞ αixi1 =limi→∞ αi(λi1xi01 + λi2xi11 + · · · + λinxin1). Thus, αi(λi1xi01 + λi2xi11 + · · · +λinxin1) < 0 for all i >> 0. Consequently, there is at least one subsequence of{αiλi1xi01}i, {αiλi2xi11}i, . . . , {αiλin+1xin1}i such that all terms are negative. Inparticular, since the coefficients αi, λij are all non-negative, it is the xij1 that willtake negative values. Moreover, the points zij lie in P or Q, therefore, we mayassume that any negative subsequence takes all of its points either from P , or fromQ, exclusively. Suppose that {α`λ`jx`j1}` is a subsequence such that α`λ`jx`j1 < 0for all ` and a fixed j. Now in R∪{∞} every sequence has a convergent subsequence,therefore, let {αmλmjxmj1}m be the convergent subsequence; xmj1 < 0 for all m(and j still fixed). Next, if we include the corresponding coordinates in Rn, thesequence {αmλmj(xmj1, xmj2, . . . , xmjn)}m has a convergent subsequence. Sincezmj ∈ P (or Q) for all m, the limit of this subsequence is necessarily in Qn

1 ; i.e., thelimit of the first coordinate is not negative. This analysis holds for any convergentsubsequence in P (or Q), contradicting the fact that x01 < 0.

In conclusion, Qn1 = 0+(conv(P ∪Q)).

Axiom 2. The operation ⊕ is associative.

Regarding (P ⊕Q)⊕R = P ⊕ (Q⊕R), we wish to prove that

conv[conv(P ∪Q) ∪R] = conv[P ∪ conv(Q ∪R)].

It suffices to show that each of these sets is equal to conv(P ∪Q∪R). Consider theset on the left. Since P ∪ Q ∪ R ⊆ conv(P ∪ Q) ∪ R ⊆ conv(P ∪ Q) ∪ R, we haveconv(P ∪Q ∪R) ⊆ conv[conv(P ∪Q) ∪R].

Conversely, as conv(P ∪Q), R ⊆ conv(P ∪Q∪R), it follows that conv(P ∪Q)∪R ⊆conv(P ∪Q∪R). Take the closed convex hull of both sides: conv[conv(P ∪Q)∪R] ⊆conv(P ∪Q∪R). This establishes that conv(P ∪Q∪R) = conv[conv(P ∪Q)∪R].Since the argument regarding the other set is analogous, this completes the proof.


Axiom 3. The operation ⊕ is commutative: this is obvious since we are takingunions of sets and order does not matter.

Axiom 4. There exists a neutral object O: this is infinity, as above.

Axiom 5. The operation � is closed; i.e., P + Q is a closed convex subset withrecession cone equal to Qn

1 .

As above, the Minkowski sum of two convex sets is again convex. Consequently,AP + AQ ⊆ A(P +Q), by Fact 1.11(8). Now since P,Q are closed, their recessioncones are equal to their asymptotic cones, thus, 0+P+0+Q ⊆ 0+(P+Q). Moreover,as 0+P = 0+Q = Qn

1 , it follows that if y ∈ 0+P\{0}, then −y /∈ 0+Q. In otherwords, 0+P and 0+Q are positively semi-independent, as per Definition 1.14. Then,by Fact 1.15, the Minkowski sum P+Q is closed, hence its recession and asymptoticcones coincide, and 0+(P +Q) ⊆ 0+P + 0+Q. Consequently, 0+(P +Q) = Qn

1 .

Axiom 6. The operation � is associative: this is obvious since addition of elementsin Rn is associative.

Axiom 7. The operation � distributes over ⊕; i.e., P�(Q⊕R) = (P�Q)⊕(P�R).

We wish to establish that P + conv(Q ∪R) = conv[(P +Q) ∪ (P +R)].

First of all, take p + z, where p ∈ P, z ∈ conv(Q ∪ R). Then z = limi→∞ zi wherezi ∈ conv(Q ∪ R). By Caratheodory’s Theorem, each zi =

∑nj=0 λijyij , where

λij ≥ 0,∑n

j=0 λij = 1, and yij ∈ Q ∪R. Therefore, we have

p+ zi = 1p+n∑

j=0

λijyij =

n∑j=0

λij

p+n∑

j=0

λijyij =n∑

j=0

λij(p+ yij).

The elements p+ yij are in P +Q or P +R, and possibly both. Therefore, the lastexpression is in conv[(P+Q)∪(P+R)]; i.e., p+zi ∈ conv[(P+Q)∪(P+R)] for eachi. Thus, limi→∞ p+zi ∈ conv[(P+Q)∪(P+R)]; i.e., p+z ∈ conv[(P+Q)∪(P+R)].Since p, z are arbitrary, we have shown that

P + conv(Q ∪R) ⊆ conv[(P +Q) ∪ (P +R)].

Conversely, since P +Q,P +R ⊆ P + conv(Q ∪R), it follows that

(P +Q) ∪ (P +R) ⊆ P + conv(Q ∪R).

Take the closed convex hull of both sides:

conv[(P +Q) ∪ (P +R)] ⊆ conv[P + conv(Q ∪R)] = conv[P + conv(Q ∪R)].

By [7, Theorem 1.1.2], conv[P +conv(Q∪R)] = P +conv(Q∪R), where both termsin the sum are closed and convex with recession cone equal to Qn

1 , as per Axiom(1). In particular, their asymptotic cones, which equal the recession cones, arepositively semi-independent, as per Definition 1.14; hence Axiom (5) applies, andP + conv(Q ∪R) is closed and convex. This means that conv[P + conv(Q ∪R)] =P + conv(Q ∪ R). Therefore, we have established the other inclusion, and henceP + conv(Q ∪R) = conv[(P +Q) ∪ (P +R)]. �


Acknowledgements

The authors thank professor Sam Lisi and Anastasiia Minenkova, a graduate stu-dent, at the University of Mississippi for helpful conversations regarding this ma-terial.

References

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McGraw-Hill Book Company, Cambridge, Massachusetts, 2001.

[5] Debreu, G., Theory of value: An axiomatic analysis of economic equilibrium, Cowles Foun-dation Monographs, 17, Yale University Press, New Haven, Connecticut, 1959.

[6] Rockafellar, R. Tyrrell, Convex Analysis, Princeton Univ. Press, Princeton, New Jersey, 1970.

[7] Schneider, Rolf, Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press,Cambridge, United Kingdom, 2013.

[8] David Speyer, Bernd Sturmfels, Tropical Mathematics, Math. Magazine, 82 (2009) no. 2,163-173.

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[10] Ziegler, G., Lectures on convex polytopes, Graduate Texts in Mathematics, Springer-Verlag,Berlin, 1994.

Department of Mathematics, University of Mississippi, Oxford, MS 38677

E-mail address: [email protected]

Department of Mathematics, University of Mississippi, Oxford, MS 38677

E-mail address: [email protected]

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