1. THE TROPICAL SEMIRING IN HIGHER DIMENSIONS
JOHN NORTON AND SANDRA SPIROFF
Abstract. We discuss the generalization, in higher dimensions, of the tropical
semiring, 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 of
their 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 defined
by
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 the
neutral object is infinity. Inspired by the survey article Tropical Mathematics by
Speyer and Sturmfels [8, p. 165], we generalize the tropical semiring to higher
dimensions. In particular, our elements are polyhedra, or more generally, closed
convex sets, in Rn
with a fixed recession cone, i.e., the directions in which the set
recedes, and the operations defined by taking the (closed) convex hull of the union
and the Minkowski sum. Indeed, when n = 1 and the recession cone is R≥0, then
this definition reduces to the original tropical semiring: the real numbers x and
y 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 vertices
at x, y. In particular, for each, the recession cone is the positive ray emanating
from the origin, or R≥0. Clearly, the union of these two sets is represented by the
inequality t ≥ min(x, y) and likewise, the Minkowski sum is given by the inequality
t ≥ 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 cone
will form a semiring. We detail this idea in our paper, considering various recession
cones. In particular, we first consider the case of bounded polyhedra, i.e., convex
polytopes, in Rn
. In this case, the recession cone is {0} and the axioms follow quite
nicely. Furthermore, we can generalize this case to that of compact (convex) sets in
Rn
. These proofs are the content of the second section1
. Prior to that, we provide
the necessary background on recession cones and asymptotic cones, and include
examples 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. 2 JOHN NORTON AND SANDRA SPIROFF
is not fixed. The main portion of the paper is dedicated to establishing the seven
axioms of the various semirings, and most especially, those dealing with the closure
of the operations. The final section of the paper considers unbounded convex sets.
We demonstrate the semirings of convex polyhedra and general convex sets, both
with recession cone equal to the positive orthant2
, where for the latter it is necessary
to 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 following
property: 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, denoted
conv S, is the set of all convex combinations of the elements of S; i.e.,
conv S = {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 recession
cone 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 if
and 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 equal
to the first quadrant Q2
1 = {(x, y) : x, y ≥ 0} of the plane.
(1) P = {(x, y) : x ≥ −5, y ≥ −18, y ≥ −5
3 x + 2};
(2) Q = {(x, y) : x ≥ −3, y ≥ −15, y ≥ −6x − 16, y ≥ −1
2 x − 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
is
one that can be expressed as the intersection of some finite collection of closed half
spaces; 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+
P
is 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 if
the only way to express x as the convex combination (1 − λ)y + λz for y, z ∈ P and
0 < 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), but
the third one is not. The finite system of inequalities associated to P is


−1 0
0 −1
−5
3 −1

 x
y
≤


5
18
−2

 ,
2The n-dimensional analogue to the first quadrant in the plane.

3. 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 recession
cone. While we only consider convex sets, this new cone is relevant since the two
definitions coincide when the convex set is closed, hence we may apply related
results 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, where
each 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. For
the converse, let z ∈ AP, and let ε > 0. Then z = limi→∞ αizi, where zi ∈ P and
αi > 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 is
some N2 ∈ N such that if i ≥ N2, then αi < 1/2. Furthermore, since zi ∈ P, there
is 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
+
1
2
· 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 (see
Fact 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 of
the operation ⊕, the following definition and result are helpful in establishing the
closure of the operation .
Definition 1.14. [5, 1.9 m., p. 22] The cones C1, C2, . . . , Ck in Rn
are positively
semi-independent if, for any ci ∈ Ci, the condition c1 + c2 + · · · + ck = 0 implies
that each ci = 0.
Fact 1.15. [3, Theorem 8] For closed and convex sets E, F ⊆ Rn
whose asymptotic
cones AE and AF are positively semi-independent, the Minkowski sum E + F is
closed and A(E + F) ⊆ AE + AF.
3This is the Euclidean metric on Rn: ρ(x, y) = ||x − y|| = [(x1 − y1)2 + · · · + (xn − yn)2]
1
2 .

4. 4 JOHN NORTON AND SANDRA SPIROFF
Example 1.16. [3, Example 2] In R2
, set E = {(x, y) : x > 0, y ≥ 1/x} and
F = {(x, y) : x < 0, y ≥ −1/x}. Note that both E and F are closed sets, but
E + F = {(x, y) : y > 0}, which is not closed.
Finally, Carath´eodory’s Theorem (see, e.g., [7, Theorem 1.1.4]) will be helpful when
considering the elements of convex sets.
Fact 1.17. [Carath´eodory’s Theorem] If a point x lies in the convex (hull of a) set
P ⊆ Rn
, then x can be written as a convex combination of no more than n+1 points
in P; i.e., there are p0, p1, . . . , pn ∈ P and λi ∈ R≥0 such that λ0 +λ1 +· · ·+λn = 1
and 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 in
Rn
. In particular, these sets are those convex polyhedra in Rn
with recession cone
equal 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 shown
below, 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 Carath´eodory’s Theorem, z =
n
i=0 λiyi, where each λi ≥ 0,
n
i=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
s
j=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. For
algorithms 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].

5. THE TROPICAL SEMIRING IN HIGHER DIMENSIONS 5
all yi ∈ Q. Thus, let m ∈ N, m < n, such that y0, . . . , ym−1 ∈ P Q and
ym, . . . , 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), and
the 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 the
set 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 taking
unions of sets and order does not matter.
Axiom 4. There exists a neutral object O such that for any convex polytope P in
Rn
, 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, and
it can be considered a convex polytope by considering ∅ as the solution set of any
inconsistent 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 hint
in [1, proof of Lemma 5.124]. Let p ∈ P and q ∈ Q. Write p =
s
j=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 =
s
j=1 λjpj +
t
k=1 µkqk = (
t
k=1 µk)
s
j=1 λjpj +(
s
j=1 λj)
t
k=1 µkqk =
s
j=1
t
k=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 sum
is 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 elements
in Rn
is associative.
Axiom 7. The operation distributes over ⊕; i.e., P (Q⊕R) = (P Q)⊕(P R).

6. 6 JOHN NORTON AND SANDRA SPIROFF
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, where
λi ≥ 0,
n
i=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 last
expression 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 last
sum are convex, the expression simply equals P + conv(Q ∪ R). Therefore, we have
established 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 generalize
the 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 are
closed 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 axioms
must 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, Corollary
5.18]), 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]). As
per [3, Corollary 11], the summation of a closed set and a compact set is closed. As
such, P Q = P + Q is closed. The fact that P + Q is bounded follows from the
fact 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-Borel
Theorem, P + Q is compact, convex in Rn
.
5This is the Euclidean norm on Rn: ||x|| = [x2
1 + · · · + x2
n]
1
2 .

7. THE TROPICAL SEMIRING IN HIGHER DIMENSIONS 7
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 polyhedra
are necessarily closed, by definition, the convex hull of two convex polyhedral sets
need not be polyhedral or closed, as evinced by Example 3.1 below; that is, if their
recession cones do not coincide. Therefore, we restrict our sets to those with the
same 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 generalization
of 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 polyhedral
nor 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 cone
equal 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 to
Qn
1 . It suffices to address the axioms regarding closure and the neutral object, since
the arguments for the remaining axioms apply here.
Axiom 1. The operation ⊕ is closed; i.e., conv(P ∪Q) is a convex polyhedron with
recession cone equal to Qn
1 .
Since conv(P ∪ Q) is convex, it remains to establish that conv(P ∪ Q) is polyhedral
with a recession cone equal to the positive orthant. Recall that, by [10, Theorem
1.2], 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)) + Qn
1 .
First of all, we consider P ∪Q, which is not necessarily convex or polyhedral, and the
directions in which it recedes. By Definition 1.6, P is the irredundant intersection
of 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 the
same holds for every other point of P. Since 0+
P = Qn
1 , the only directions in
which P recedes are {(x1, . . . , xn) : xi ≥ 0 for all i}. In particular, P recedes in
the direction of each ei = (0, . . . , 0, 1, 0, . . . , 0), the i-th standard basis vector. This
means 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 some
ci ∈ 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 bounded
by 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 is
obvious. Therefore, noting that 0+
(conv(P ∪ Q)) = Qn
1 , as per (3.1), we may write
conv(P ∪ Q) = conv(ext(P) ∪ ext(Q)) + Qn
1 , and the latter, by [10, Theorem 1.2],
is polyhedral.

8. 8 JOHN NORTON AND SANDRA SPIROFF
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 of
mix − 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))+ Q2
1, and in particular,
that conv(P ∪ Q) is polyhedral.
Axiom 4. There exists a neutral object O ⊆ Rn
∪{∞} such that for all polyhedral
convex 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 is
infinity, since P recedes in all directions of the positive orthant.
Axiom 5. The operation is closed; i.e., P + Q is a convex polyhedron with
recession cone equal to Qn
1 .
By [6, Corollary 19.3.2], the Minkowski sum of two polyhedral convex sets in Rn
is
polyhedral, and it is convex (see, e.g., [7, Theorem 1.1.2]). Therefore, it remains to
show that 0+
(P +Q) = Qn
1 . Since polyhedral convex sets are closed, their recession
cones 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.

9. THE TROPICAL SEMIRING IN HIGHER DIMENSIONS 9
3.2. The Semiring of Closed Convex Sets. In this section, we generalize the
above work with convex polyhedra to general closed convex subsets of Rn
with
recession cone equal to the positive orthant. As evinced in Example 1.13, pathology
arises if the sets are not assumed to be closed; however, despite taking two convex
sets that are closed, neither the convex hull of the union nor the Minkowski sum
need be closed, as per Examples 3.1 and 1.16, respectively. Therefore, in this section
we take our definition of ⊕ to be the closed convex hull of the union and make use of
the material at the end of Section 1 in order to ensure the closure of the Minkowski
sum. In particular, we have:
Theorem 3.3. The set of all closed convex sets in Rn
∪ {∞} with recession cone
equal 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 the
positive orthant, denoted Qn
1 .
Axiom 1. The operation ⊕ is closed; i.e., conv(P ∪ Q) is a closed convex subset
with recession cone equal to Qn
1 .
First of all, as ⊕ is defined as the closed convex hull of the union, it only remains
to 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 Qn
1 ⊆ 0+
(conv(P ∪ Q)) and A(conv(P ∪ Q)) ⊆ Qn
1 .
For the right inclusion, take w ∈ Qn
1 , and let z ∈ conv(P ∪ Q). By Carath´eodory’s
Theorem, z can be expressed as a convex combination of exactly n + 1 elements
from 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 any
case, 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 as
a 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.

10. 10 JOHN NORTON AND SANDRA SPIROFF
For a fixed j, consider the sequence {αi(λijzij)}i. Since the αi are non-negative
real numbers converging to 0 and 0 ≤ λij ≤ 1, the coefficients αiλij also converge
to 0 from above. Thus, as infinitely many zij are either in P or in Q, it follows
that limi→∞(αiλij)zij, if it exists, is an element of Qn
1 since AP = 0+
P and
AQ = 0+
Q, by Fact 1.11(7). Assume, for the moment, that all such limits exist,
and set rj = limi→∞(αiλij)zij. Then
z = lim
i→∞
αizi = lim
i→∞
αi


n
j=0
λijzij

 =
n
j=0
lim
i→∞
αiλijzij =
n
j=0
rj ∈ Qn
1 ,
and A(conv(P ∪ Q)) ⊆ Qn
1 , as desired.
However, as it can not simply be assumed that all of the above limits exist, at this
point we take a different tack. Write z = (x01, . . . , x0n), zi = (xi1, . . . , xin), and
zij = (xij1, . . . , xijn), so that equation (3.2) can be rewritten as
(3.3) (x01, . . . , x0n) = lim
i→∞
αi(xi1, . . . , xin) = lim
i→∞
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. In
particular, since the coefficients αi, λij are all non-negative, it is the xij1 that will
take negative values. Moreover, the points zij lie in P or Q, therefore, we may
assume that any negative subsequence takes all of its points either from P, or from
Q, exclusively. Suppose that {α λ jx j1} is a subsequence such that α λ jx j1 < 0
for 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
, the
sequence {αmλmj(xmj1, xmj2, . . . , xmjn)}m has a convergent subsequence. Since
zmj ∈ P (or Q) for all m, the limit of this subsequence is necessarily in Qn
1 ; i.e., the
limit of the first coordinate is not negative. This analysis holds for any convergent
subsequence in P (or Q), contradicting the fact that x01 < 0.
In conclusion, Qn
1 = 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 the
set on the left. Since P ∪ Q ∪ R ⊆ conv(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 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.

11. THE TROPICAL SEMIRING IN HIGHER DIMENSIONS 11
Axiom 3. The operation ⊕ is commutative: this is obvious since we are taking
unions 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 with
recession 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 recession
cones 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 other
words, 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 asymptotic
cones 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 elements
in 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 where
zi ∈ conv(Q ∪ R). By Carath´eodory’s Theorem, each zi =
n
j=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 last
expression is in conv[(P +Q)∪(P +R)]; i.e., p+zi ∈ conv[(P +Q)∪(P +R)] for each
i. 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 terms
in 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, are
positively semi-independent, as per Definition 1.14; hence Axiom (5) applies, and
P + 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 hence
P + conv(Q ∪ R) = conv[(P + Q) ∪ (P + R)].

12. 12 JOHN NORTON AND SANDRA SPIROFF
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.
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Department of Mathematics, University of Mississippi, Oxford, MS 38677
E-mail address: jmnorton@go.olemiss.edu
Department of Mathematics, University of Mississippi, Oxford, MS 38677
E-mail address: spiroff@olemiss.edu