Tropical convexity

Dans le document The DART-Europe E-theses Portal (Page 86-90)

In this section, we recall some basic facts about convexity in the usual and tropical sense. The convex hull of a set X Kn, denoted conv(X), can be defined as the smallest (inclusionwise) convex set that contains X. This set is characterized by Carathéodory’s theorem.

Theorem 5.7 (Carathéodory’s theorem, [Sch87, Corollary 7.1j]). IfX Kn, then we have the equality

conv(X) = {n+1

k=1

λkx(k)Kn:∀k,x(k)X ∧ ∀k,λ⩾0 n+1

k=1

λk= 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 Kn is a semialgebraic set, then conv(S) is also semialgebraic.

Proof. By Theorem 5.7,conv(S)is definable inLor. 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 setX⊂Tn istropically convex if for everyx, y∈X and every λ, µ∈Tsuch thatλ⊕µ= 0 the point(λ⊙x)⊕⊙y) belongs toX. We say that X⊂Tn is a tropical (convex) cone if(λ⊙x)⊕⊙y)∈X for all λ, µ∈T.

Remark 5.11. The quantity(λ⊙x)⊕⊙y) forλ⊕µ= 0corresponds to the tropical analogue of a convex combination ofxandy. Indeed, the scalarsλandµare 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 allx, y∈Tn,λ, µ∈T, and alli, j∈[m]such thatij we haveQij(⊙x)⊕⊙y))= (λ⊙Qij(x))⊕(µ⊙Qij(y))and the same is true forQ+ii. Hence, ifx, y∈ S(Q(1), . . . , Q(n)), then

Q+ii(⊙x)⊕⊙y))= (λ⊙Q+ii(x))⊙Q+ii(y))

⩾(λ⊙Qii(x))⊙Qii(y)) =Qii(⊙x)⊕⊙y)) for all i∈[m]and

Q+ii(⊙x)⊕⊙y))⊙Q+jj(⊙x)⊕⊙y))

= (

⊙Q+ii(x))⊙Q+ii(y))

)(⊙Q+jj(x))⊙Q+jj(y)) )

(λ2⊙Q+ii(x)⊙Q+jj(x))(µ2⊙Q+ii(y)⊙Q+jj(y))

⩾(λ⊙Qij(x))2⊙Qij(x))2

=((λ⊙Qij(x))(µ⊙Qij(x)))2

= (

Qij(⊙x)⊕⊙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 setX Tnwe can define itstropical convex hull, denotedtconv(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 Tn, then we have the equality tconv(X) =

{n+1

k=1

k⊙x(k)) :∀k, x(k)∈X

n+1

k=1

λk = 0 }

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

Yp = {p

k=1

k⊙x(k)) : ∀k, x(k)∈X

p k=1

λk= 0 }

.

An easy induction shows that Yp tconv(X) for all p ⩾ 0. Moreover, the set p=1Yp is tropically convex and hence tconv(X) = p=1Yp. Suppose that y Yp for some p ⩾ 1. For every coordinate l [n] we can find k(l) [p] such that yl = λk(l)⊙xk(l)l . Moreover, there exists k [p]such that λk = 0. Hencey =(⊕pl=1k(l)⊙xk(l)))⊕x(k), which implies that y∈Yn+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 Tn are tropically convex. Then, we have the equality tconv(X∪Y) ={⊙x)⊕⊙y)∈Tn: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 containsX andY 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 Kn is a convex set, thenval(X) is tropically convex.

5.1. Tropical convexity 87 Proof. Letx, y∈val(X)and take anyλ, µ∈Tsuch thatλ⊕µ= 0. Without loss of generality, suppose that λ= 0. Take any points xX val1(x) and y X val1(y). Let us look at two cases. Ifµ <0, then for any real positive constantc, we have1−ctµ>0, and so the point z= (1−ctµ)x+ctµybelongs toX. Hence, if we choosecsuch thatc̸=−lc(xk)/lc(yk)for all k∈[n]satisfyingyk̸= 0, thenval(z) = (λ⊙x)⊕⊙y). Ifµ= 0, then we take a real constant c∈]0,1[ such that for allk∈[n]satisfying yk ̸= 0we have c/(1−c)̸=lc(xk)/lc(yk). Then, the point z= (1−c)x+cy belongs to X and we have val(z) = (λ⊙x)⊕⊙y).

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

Lemma 5.19. If X Kn0 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 existλ1, . . . ,λn+1⩾0 andx(1), . . . ,x(n+1) X such thaty=λ1x(1)+· · ·+λn+1x(n+1). Hence, using the fact that X Kn0, we have

val(y) =(val(λ1)val(x1))⊕ · · · ⊕(val(λn+1)val(xn+1)).

Furthermore, we haven+1k=1λk= 1 and hencen+1k=1val(λk) = 0. Therefore,val(y)tconv(X) by Theorem 5.15. Conversely, take any point y tconv(X). By Theorem 5.15, we can find λ1, . . . , λn+1 T,n+1k=1λk= 0and x(1), . . . , x(n+1)∈X such thaty = (λ1⊙x(1))⊕. . .n+1 x(n+1)). We define λk := tλk/(n+1l=1 tλl). Observe that for all k, val(λk) = λk because the termn+1l=1 tλl has valuation n+1l=1 λl= 0. Moreover, we have λk ⩾0 and n+1k=1λk = 1. Take any points x(1), . . . ,x(n+1) X such that val(x(k)) = x(k) for all k [n+ 1]. The point y=λ1x1+· · ·+λn+1xn+1 belongs toconv(X) and verifies val(y) =y.

Example 5.20. The assumption thatX 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 setSTn is a tropicalization of a convex semialgebraic set if and only if Sis tropically convex and every stratum of Sis 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 Kn0 included in the nonnegative orthant of Kn and such that val(S) =S. The set conv(S) is semialgebraic by Lemma 5.9 and satisfiesval(conv(S)) = tconv(S) =S by Lemma 5.19.

Remark 5.22. The considerations of this section extend to the case of tropicalization of convex cones. Indeed, if X Tn 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

k⊙x(k)) :∀k, x(k)∈X, λkT}.

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 Kn is a convex cone, then val(X) is a tropical cone. Indeed, if we take x, y val(X), λ, µ T, and x,yX such that val(x) =x, val(y) =y, then there is c >0 such that z:=tλx+ctµyX and val(z) = (λ⊙x)⊕⊙y) (we choose c such that c ̸= lc(xk)/lc(yk) for all k [n]

satisfying yk ̸= 0). Moreover, ifX Kn0 is any set included in the nonnegative orthant and cone(·) denotes the conic hull in Kn, 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 Tn is a tropicalization of a convex semialgebraic cone if and only if S is a tropical cone that has closed semilinear strata.

5.2 Real tropical cones as sublevel sets of dynamic programming

Dans le document The DART-Europe E-theses Portal (Page 86-90)