In this section, we recall some basic facts about convexity in the usual and tropical sense. The
convex hull of a set **X***⊂*K* ^{n}*, denoted conv(X), can be defined as the smallest (inclusionwise)
convex set that contains

*. This set is characterized by Carathéodory’s theorem.*

**X****Theorem 5.7** (Carathéodory’s theorem, [Sch87, Corollary 7.1j]). *If X*

*⊂*K

^{n}*, then we have the*

*equality*

conv(X) =
{* ^{n+1}*∑

*k=1*

**λ***k***x**^{(k)}*∈*K* ^{n}*:

*∀k,*

**x**^{(k)}

*∈*

**X***∧ ∀k,*⩾0

**λ***∧*

^{n+1}^{∑}

*k=1*

**λ***k*= 1
}

*.*

5.1. Tropical convexity 85
*Remark* 5.8. We point out that the proof of Theorem 5.7 given in [Sch87, Corollary 7.1j] is valid
over every ordered field.

As a corollary, we obtain that the class of semialgebraic sets is closed under taking convex hulls.

**Lemma 5.9.** *If* **S***⊂*K^{n}*is a semialgebraic set, then* conv(* S*)

*is also semialgebraic.*

*Proof.* By Theorem 5.7,conv(* S*)is definable in

*L*or. Hence, it is semialgebraic by Lemma 2.111.

Let us now move to tropical convexity, referring the reader to [CGQ04, DS04] for more information.

**Definition 5.10.** We say that a set*X⊂*T* ^{n}* is

*tropically convex*if for every

*x, y∈X*and every

*λ, µ∈*Tsuch that

*λ⊕µ*= 0 the point(λ

*⊙x)⊕*(µ

*⊙y)*belongs to

*X. We say that*

*X⊂*T

*is a*

^{n}*tropical (convex) cone*if(λ

*⊙x)⊕*(µ

*⊙y)∈X*for all

*λ, µ∈*T.

*Remark* 5.11. The quantity(λ*⊙x)⊕*(µ*⊙y)* for*λ⊕µ*= 0corresponds to the tropical analogue
of a convex combination of*x*and*y. Indeed, the scalarsλ*and*µ*are implicitly “nonnegative” in
the tropical sense, as they are greater than or equal to the tropical zero element *−∞*. Besides,
their tropical sum equals the tropical unit0.

*Example* 5.12. Any tropical Metzler spectrahedral cone is a tropical cone. Indeed, if we use the
notation as in Definition 4.13, then for all*x, y∈*T* ^{n}*,

*λ, µ∈*T, and all

*i, j∈*[m]such that

*i*⩽

*j*we have

*Q*

^{−}

_{ij}^{(}(λ

*⊙x)⊕*(µ

*⊙y)*

^{)}= (λ

*⊙Q*

^{−}*(x))*

_{ij}*⊕(µ⊙Q*

^{−}*(y))and the same is true for*

_{ij}*Q*

^{+}

*. Hence, if*

_{ii}*x, y∈ S*(Q

^{(1)}

*, . . . , Q*

^{(n)}), then

*Q*^{+}_{ii}^{(}(λ*⊙x)⊕*(µ*⊙y)*^{)}= (λ*⊙Q*^{+}* _{ii}*(x))

*⊕*(µ

*⊙Q*

^{+}

*(y))*

_{ii}⩾(λ*⊙Q*^{−}* _{ii}*(x))

*⊕*(µ

*⊙Q*

^{−}*(y)) =*

_{ii}*Q*

^{−}

_{ii}^{(}(λ

*⊙x)⊕*(µ

*⊙y)*

^{)}for all

*i∈*[m]and

*Q*^{+}_{ii}^{(}(λ*⊙x)⊕*(µ*⊙y)*^{)}*⊙Q*^{+}_{jj}^{(}(λ*⊙x)⊕*(µ*⊙y)*^{)}

= (

(λ*⊙Q*^{+}* _{ii}*(x))

*⊕*(µ

*⊙Q*

^{+}

*(y))*

_{ii})*⊙*^{(}(λ*⊙Q*^{+}* _{jj}*(x))

*⊕*(µ

*⊙Q*

^{+}

*(y)) )*

_{jj}⩾^{(}*λ*^{⊙}^{2}*⊙Q*^{+}* _{ii}*(x)

*⊙Q*

^{+}

*(x)*

_{jj}^{)}

*⊕*

^{(}

*µ*

^{⊙}^{2}

*⊙Q*

^{+}

*(y)*

_{ii}*⊙Q*

^{+}

*(y)*

_{jj}^{)}

⩾(λ*⊙Q**ij*(x))^{⊙}^{2}*⊕*(µ*⊙Q**ij*(x))^{⊙}^{2}

=^{((}*λ⊙Q** _{ij}*(x)

^{)}

*⊕*

^{(}

*µ⊙Q*

*(x)*

_{ij}^{))}

^{⊙}^{2}

= (

*Q*^{−}_{ij}^{(}(λ*⊙x)⊕*(µ*⊙y)*^{))}^{⊙}^{2}

for all *i < j*. A more abstract proof of the same fact can be done using Lemma 5.18 below
combined with Proposition 4.14.

**Definition 5.13.** For any set*X* *⊂*T* ^{n}*we can define its

*tropical convex hull, denoted*tconv(X), as the smallest (inclusionwise) tropically convex set that contains

*X.*

*Remark* 5.14. The tropical convex hull is well defined because the intersection of any number
of tropically convex sets is tropically convex.

86 Chapter 5. Tropical analogue of the Helton–Nie conjecture

Figure 5.1: Tropical convex hull of three points.

Carathéodory’s theorem is still true in the tropical setting:

**Theorem 5.15** ([Hel88], [BH04], [DS04]). *If* *X* *⊂*T^{n}*, then we have the equality*
tconv(X) =

{* ^{n+1}*⊕

*k=1*

(λ*k**⊙x*^{(k)}) :*∀k, x*^{(k)}*∈X* *∧*

*n+1*⊕

*k=1*

*λ**k* = 0
}

*.*
*Sketch of the proof.* For every *p*⩾1 we denote

*Y**p* =
{⊕^{p}

*k=1*

(λ*k**⊙x*^{(k)}) : *∀k, x*^{(k)}*∈X* *∧*

⊕*p*
*k=1*

*λ**k*= 0
}

*.*

An easy induction shows that *Y*_{p}*⊂* tconv(X) for all *p* ⩾ 0. Moreover, the set ^{∪}^{∞}_{p=1}*Y** _{p}* is
tropically convex and hence tconv(X) =

^{∪}

^{∞}

_{p=1}*Y*

*. Suppose that*

_{p}*y*

*∈*

*Y*

*for some*

_{p}*p*⩾ 1. For every coordinate

*l*

*∈*[n] we can find

*k(l)*

*∈*[p] such that

*y*

*=*

_{l}*λ*

_{k(l)}*⊙x*

^{k(l)}*. Moreover, there exists*

_{l}*k*

^{∗}*∈*[p]such that

*λ*

*k*

*= 0. Hence*

^{∗}*y*=

^{(⊕}

^{p}*(λ*

_{l=1}

_{k(l)}*⊙x*

*)*

^{k(l)}^{)}

*⊕x*

^{(k}

^{∗}^{)}, which implies that

*y∈Y*

*n+1*.

*Example* 5.16. Figure 5.1 depicts a tropical convex hull of three points: (1,5),(3,2), and(8,7).

Note that the point (4,4)can be written as(4,4) =^{(}(*−*4)*⊙*(8,7)^{)}*⊕*^{(}(*−*1)*⊙*(1,5)^{)}*⊕*(3,2).

We also need the following lemma.

**Lemma 5.17.** *Suppose that sets* *X, Y* *⊂*T^{n}*are tropically convex. Then, we have the equality*
tconv(X*∪Y*) =*{*(λ*⊙x)⊕*(µ*⊙y)∈*T* ^{n}*:

*x∈X, y∈Y, λ⊕µ*= 0

*}.*

*Sketch of the proof.* The inclusion *⊃*follows immediately from the definition of tropical convex
hull. The other inclusion holds because the set on the right-hand side contains*X* and*Y* and is
tropically convex.

A relation between the convexity in Kand the tropical convexity is shown in the next two lemmas.

**Lemma 5.18.** *If* **X***⊂*K^{n}*is a convex set, then*val(X) *is tropically convex.*

5.1. Tropical convexity 87
*Proof.* Let*x, y∈*val(X)and take any*λ, µ∈*Tsuch that*λ⊕µ*= 0. Without loss of generality,
suppose that *λ*= 0. Take any points **x**∈**X***∩*val^{−}^{1}(x) and **y***∈ X*

*∩*val

^{−}^{1}(y). Let us look at two cases. If

*µ <*0, then for any real positive constant

*c, we have*1

*−ct*

^{µ}*>*0, and so the point

*= (1*

**z***−ct*

*)x+*

^{µ}*ct*

^{µ}*belongs to*

**y***. Hence, if we choose*

**X***c*such that

*c̸=−*lc(x

*k*)/lc(y

*k*)for all

*k∈*[n]satisfying

**y**

_{k}*̸*= 0, thenval(z) = (λ

*⊙x)⊕*(µ

*⊙y). Ifµ*= 0, then we take a real constant

*c∈*]0,1[ such that for all

*k∈*[n]satisfying

**y**

_{k}*̸*= 0we have

*c/(1−c)̸*=

*−*lc(x

*)/lc(y*

_{k}*). Then, the point*

_{k}*= (1*

**z***−c)x*+

*cy*belongs to

*and we have val(z) = (λ*

**X***⊙x)⊕*(µ

*⊙y).*

The next lemma shows that a tighter relation holds for sets included in the nonnegative orthant ofK.

**Lemma 5.19.** *If* **X***⊂*K^{n}_{⩾}_{0} *is any set, then we have* val(conv(X)) = tconv(val(X)).

*Proof.* We start by proving the inclusion *⊂*. Take a point **y***∈* conv(X). By Theorem 5.7,
there exist**λ**_{1}*, . . . , λ*

*⩾0 and*

_{n+1}

**x**^{(1)}

*, . . . ,*

**x**^{(n+1)}

*∈*such that

**X***=*

**y**

**λ**_{1}

**x**^{(1)}+

*· · ·*+

**λ**

_{n+1}

**x**^{(n+1)}. Hence, using the fact that

**X***⊂*K

^{n}_{⩾}

_{0}, we have

val(y) =^{(}val(λ_{1})*⊙*val(x_{1})^{)}*⊕ · · · ⊕*^{(}val(λ* _{n+1}*)

*⊙*val(x

*)*

_{n+1}^{)}

*.*

Furthermore, we have^{∑}^{n+1}_{k=1}**λ*** _{k}*= 1 and hence

^{⊕}

^{n+1}*val(λ*

_{k=1}*) = 0. Therefore,val(y)*

_{k}*∈*tconv(X) by Theorem 5.15. Conversely, take any point

*y*

*∈*tconv(X). By Theorem 5.15, we can find

*λ*1

*, . . . , λ*

*n+1*

*∈*T,

^{⊕}

^{n+1}

_{k=1}*λ*

*k*= 0and

*x*

^{(1)}

*, . . . , x*

^{(n+1)}

*∈X*such that

*y*= (λ1

*⊙x*

^{(1)})

*⊕. . .*(λ

*n+1*

*⊙*

*x*

^{(n+1)}). We define

**λ***:=*

_{k}*t*

^{λ}

^{k}*/(*

^{∑}

^{n+1}

_{l=1}*t*

^{λ}*). Observe that for all*

^{l}*k,*val(λ

*) =*

_{k}*λ*

*because the term*

_{k}^{∑}

^{n+1}

_{l=1}*t*

^{λ}*has valuation*

^{l}^{⊕}

^{n+1}

_{l=1}*λ*

*= 0. Moreover, we have*

_{l}

**λ***⩾0 and*

_{k}^{∑}

^{n+1}

_{k=1}

**λ***= 1. Take any points*

_{k}

**x**^{(1)}

*, . . . ,*

**x**^{(n+1)}

*∈*

*such that val(x*

**X**^{(k)}) =

*x*

^{(k)}for all

*k*

*∈*[n+ 1]. The point

*=*

**y***1*

**λ***1+*

**x***· · ·*+

**λ***n+1*

**x***n+1*belongs toconv(X) and verifies val(y) =

*y.*

*Example* 5.20. The assumption that* X* lies in the nonnegative orthant cannot be omitted. To
see this, take

*=*

**X***{−*1,1

*} ⊂*K. We have val(X) =

*{*0

*}*= tconv(val(X)), but val(conv(X)) = val([

*−*1,1]) = [

*−∞,*0].

The last two lemmas, together with Theorem 3.1, allow us to give our first characterization of tropicalizations of convex semialgebraic sets.

**Proposition 5.21.** *A set*S_{⊂}_{T}^{n}*is a tropicalization of a convex semialgebraic set if and only*
*if* S*is tropically convex and every stratum of* S*is a closed semilinear set.*

*Proof.* The “only if” part follows from Theorem 3.1 and Lemma 5.18. To prove the
oppo-site implication, suppose that S is tropically convex and has closed semilinear strata. Then,
Lemma 3.13 shows that there exists a semialgebraic set **S***⊂*K^{n}_{⩾}_{0} included in the nonnegative
orthant of K* ^{n}* and such that val(

*) =S. The set conv(*

**S***) is semialgebraic by Lemma 5.9 and satisfiesval(conv(*

**S***)) = tconv(S) =S by Lemma 5.19.*

**S***Remark* 5.22. The considerations of this section extend to the case of tropicalization of convex
cones. Indeed, if *X* *⊂*T* ^{n}* is any set, the we can define its

*tropical conic hull,*tcone(X) as the smallest tropical cone that contains

*X. In this case, the tropical Carathéodory theorem states*that we have the equality

tcone(X) =
{⊕^{n}

*k=1*

(λ*k**⊙x*^{(k)}) :*∀k, x*^{(k)}*∈X, λ**k**∈*T^{}}*.*

88 Chapter 5. Tropical analogue of the Helton–Nie conjecture
The analogues of the other propositions stated above are also true for cones. If **X***⊂*K* ^{n}* is a
convex cone, then val(X) is a tropical cone. Indeed, if we take

*x, y*

*∈*val(X),

*λ, µ*

*∈*T, and

*such that val(x) =*

**x,****y**∈**X***x,*val(y) =

*y, then there is*

*c >*0 such that

*:=*

**z***t*

^{λ}*+*

**x***ct*

^{µ}*and val(z) = (λ*

**y**∈**X***⊙x)⊕*(µ

*⊙y)*(we choose

*c*such that

*c*

*̸=*

*−*lc(x

*k*)/lc(y

*k*) for all

*k*

*∈*[n]

satisfying **y**_{k}*̸*= 0). Moreover, if**X***⊂*K^{n}_{⩾}0 is any set included in the nonnegative orthant and
cone(*·*) denotes the conic hull in K* ^{n}*, then we have val(cone(X)) = tcone(val(X)). The proof
of this fact is done as in Lemma 5.19, replacing the Carathéodory’s theorem by its analogue
for cones [Sch87, Corollary 7.1i]. In particular, by repeating the proof of Proposition 5.21, we
get that a set S

_{⊂}_{T}

*is a tropicalization of a convex semialgebraic cone if and only if S is a tropical cone that has closed semilinear strata.*

^{n}