Observe that a shortest path from vertexito vertexj, which uses at mostr≥2 edges, consists of a shortest path from vertex i to some vertexk, which uses at most r−1 edges, and the edge (k, j).
That is, the shortest paths from vertexito vertexj satisfy the equation
d(r)ij = min{d(r−1)ik +dkj|1≤k≤n}, 2≤r≤n−1. (3.3) The tropicalization of this equation yields
d(r)ij = Mn k=1
d(r−1)ik ⊙dkj, 2≤r≤n−1. (3.4)
The right-hand side is the tropical product of theith row of DG⊙r−1 and the jth column ofDG. Thus the left-hand side is the (i, j)th entry of the matrixDG⊙r. Hence, the assertion follows. ⊓⊔ The iterative evaluation of Eq. (3.3) is known as Floyd-Warshall algorithm for finding the shortest paths between each pair of vertices in a digraph.
Example 3.5.Consider the following digraphG,
?>=<
and the tropical matrix products are
D⊙2G =
We consider the Euclideann-spaceRn equipped with the ordinary scalar product
hu, vi=u1v1+· · ·+unvn, u, v∈Rn. (3.5) The Euclidean distance between two pointsuandv in Rn is defined as
ku−vk=p
hu−v, u−vi. (3.6)
A setC inRn is called convex if it contains the line segment connecting any two points inC. The line segment between two pointsuandv in Rn is given as
[u, v] ={λu+ (1−λ)v|0≤λ≤1}. (3.7) Simple examples of convex sets are the singleton sets{v}, where v∈Rn, and the Euclidean spaceRn. There is a simple way to construct new convex sets from given ones.
Proposition 3.6.The intersection of an arbitrary collection of convex sets is convex.
Proof. If a line segment belongs to every set in the collection, it also belongs to the intersection. ⊓⊔ If a set is not itself convex, itsconvex hull is the smallest convex set containing it (Fig. 3.2). The convex hull of a setS in Rn is denoted by conv(S).
Fig. 3.2.A set and its convex hull, a convex polygon.
Proposition 3.7.If S is a subset of Rn, then its convex hull is
conv(S) ={λ1s1+. . .+λmsm|m≥0, si∈S, λi≥0, X
3.3 Geometric Zoo 53
LetCbe a convex set inRncontainingS. We show that conv(S) lies inCby using induction on the size (i.e., number of terms) of the linear combinations. Each element of S lies inC. Lets1, . . . , sm+1
be elements ofS. Consider the convex combination
s=λ1s1+. . .+λmsm+λm+1sm+1, whereλi ≥0, 1≤i≤m+ 1, andP
iλi= 1. Ifλm+1 = 1, thens=sm+1∈C, and ifλm+1 = 0, then by induction we haves∈C. Otherwise, we have
s= (1−λm+1)
Thus, by induction, the convex combination s′=
belongs toC, as required. The remaining assertions are left to the reader. ⊓⊔ A linear combinations of the formλ1s1+. . .+λmsm, where si∈S,λi≥0, andP
iλi= 1, is called a convex combination.
A polytope is the convex hull of a finite set in Rn. If the set is S = {s1, . . . , sm} in Rn, then by Prop. 3.7, the corresponding polytope can be expressed as
conv(S) ={λ1s1+. . .+λmsm|λi≥0,X
i
λi= 1}. (3.9)
In lower dimensions, polytopes are familiar geometric figures: A polytope in R is a line segment, a polytope inR2 is a line segment or aconvex polygon(Fig. 3.2), and a polytope inR3 is a line segment, a convex polygon lying in a plane, or aconvex polyhedron. In particular, alattice polytopeis a polytope given by the convex hull of a set of integral points.
Example 3.8.The mathematical softwarepolymakewas designed to work with polytopes. Each poly-tope inpolymakeis treated as an object and is given by a file storing the data. The programpolymake allows to construct polytopes from scratch or by applying constructions to existing polytopes.
Consider the lattice polytope given by the convex hull of the points (0,8), (0,7), (0,6), (0,5), (1,6), (1,5), (1,4), (1,3), (2,4), (2,3), and (3,2) (Fig. 5.7). Inpolymake, this polytope can be specified in a text file, saydude, containing the following information
POINTS 1 0 8 1 0 7
1 0 6
The points are always represented in homogeneous coordinates, where the first coordinate is used for
homogenization. ♦
In particular, an-dimensional simplex orn-simplex is the convex hull ofn+ 1 pointsm1, . . . , mn+1
inRnsuch that the vectorsm2−m1, . . . , mn+1−m1form a basis ofRn. An-simplex can be constructed from a (n−1)-simplex inRn−1 by adding one point in the n-th dimension and connecting the point with all points of the (n−1)-simplex. In this way, one obtains inductively simplices that are singleton points, line segments, triangles, tetrahedrons (Fig. 3.3), and so on.
✎✎✎✎✎✎✎✎✎✎✎✎✎✎
Each polytope has a well-defined dimension. To see this, we need to develop the theory of affine subspaces. Anaffine subspaceofRnis a subsetAofRnwith the property that ifm≥0 ands1, . . . , sm∈ Athen λ1s1+. . .+λmsm∈A withλ1, . . . , λm∈R, whenever P
iλi= 1. Linear combinations of the formλ1s1+. . .+λmsm, wheresi∈A andP
iλi= 1, are calledaffine combinations.
Given a subsetS ofRn and a vectorv∈Rn, thetranslate ofS by v is the set
v+S ={v+s|s∈S}. (3.10)
Proposition 3.9.Each affine subspace of Rn is a translate of a unique linear subspace of Rn. Proof. LetAbe an affine subspace ofRn andv∈A. Consider the translate
−v+A={λ1a1+. . .+λmam|m≥0, ai ∈A,X
i
λi= 0}.
It is easy to check that the translate −v+A is a linear subspace of Rn. Since A = v+ (−v+A), it follows that Ais a translate of the linear subspace −v+A. Moreover, if v, w ∈A, then the above
3.3 Geometric Zoo 55
representation of a translate shows that the linear subspaces−v+Aand−w+Aare equal. It follows that the affine subspaceA is a translate of a unique linear subspace ofRn. ⊓⊔ The dimension of an affine subspace in Rn is defined as the dimension of the linear subspace of Rn corresponding to it as in Prop. 3.9.
Proposition 3.10.The translate of a polytope is a polytope.
Proof. LetP be a polytope inRnand letv∈Rn. By definition, there is a subsetS={s1, . . . , sm}ofRn such thatP = conv(S). Claim thatv+ conv(S) = conv(v+S). Indeed, letw∈P. Writew=P
iλisi, whereλi≥0, 1≤i≤m, andP
iλi= 1. Then v+X
i
λisi =X
i
λi(v+si).
The left-hand side is a point in v+ conv(S) and the right-hand side is a point in conv(v+S). This proves the claim. Thus the translate v+P is the convex hull of the set v+S and hence v+P is a
polytope. ⊓⊔
If a set is not itself an affine subspace ofRn, itsaffine hull is the smallest affine subspace containing it. The affine hull of a setS inRn is denoted by aff(S).
Proposition 3.11.IfS is a subset of Rn, then
aff(S) ={λ1s1+. . .+λmsm|m≥0, si∈S, λi ∈R, X
i
λi= 1}. (3.11) Moreover,
• for each subset S of Rn,aff(aff(S)) = aff(S).
• If S1 andS2 are subsets of Rn such that S1⊆S2, thenaff(S1)⊆aff(S2).
Proof. By definition, the set aff(S) is an affine subspace ofRn. Moreover, for eachs∈S,s= 1·s∈aff(S) and thus aff(S) contains the setS.
LetAbe an affine subspace ofRncontainingS. We show that aff(S) lies inAby using induction on the size (i.e, number of terms) of the linear combinations. Each element ofSlies inA. Lets1, . . . , sm+1
be elements ofS. Consider the affine combination
s=λ1s1+. . .+λmsm+λm+1sm+1, whereP
iλi= 1. Ifλm+1= 1, thens=sm+1∈A, and ifλm+1= 0, then by induction we haves∈A.
Otherwise, we have
s= (1−λm+1) Xm i=1
λi
1−λm+1
si+λm+1sm+1.
and Xm
i=1
λi
1−λm+1
= 1.
Thus, by induction, the affine combination
s′= Xm i=1
λi
1−λm+1
si
belongs toA. SinceAis an affine subspace and containsS, it follows that the elements= (1−λm+1)s′+ λm+1sm+1 belongs toA, as required. The remaining assertions are left to the reader. ⊓⊔ For instance, the affine hull of a point is a point, the affine hull of a line segment is a line, the affine hull of a convex polygon is a plane, and the affine hull of a convex polyhedron is the Euclidean 3-space.
Thedimension of a polytope inRn is defined as the dimension of its affine hull. This is the small-est affine space containing the polytope. For instance, a point has dimension 0, a line segment has dimension 1, a convex polygon has dimension 2, and a convex polyhedron has dimension 3.
Proposition 3.12.Eachn-simplex in Rn has dimensionn.
Proof. LetC be an-simplex inRn given by the pointsm1, . . . , mn+1. The affine hull ofC contains the pointsm2−m1, . . . , mn+1−m1. These points form a basis ofRnand thus the affine hull has dimension at leastn. But aff(C) is an affine subspace ofRn and so has dimension at mostn. ⊓⊔ The unbounded counterparts of polytopes are cones. A cone in Rn is a subset C of Rn with the property that ifm≥0 ands1, . . . , sm∈C thenλ1s1+. . .+λmsm, wheneverλi≥0, 1≤i≤m.
If a set is not itself a cone, itspositive hull is the smallest cone containing it. The positive hull of a setS in Rn is denoted by pos(S).
Proposition 3.13.IfS is a subset of Rn, then the positive hull of S is
pos(S) ={λ1s1+. . .+λmsm|m≥0, si∈S, λi≥0}. (3.12) Proof. Letu, v ∈S. By definition, the set pos(S) is a cone inRn. Moreover, for eachs∈S,s= 1·s∈ pos(S) and thus pos(S) contains the setS.
Finally, letC be a cone inRn containingS. We show that pos(S) lies inC by using induction on the size (i.e., number of terms) of the linear combinations. Each element ofSlies inC. Lets1, . . . , sm+1
be elements ofS. Consider the linear combination
s=λ1s1+. . .+λmsm+λm+1sm+1,
whereλi≥0, 1≤i≤m+ 1. Ifλm+1= 0, then by induction we haves∈C. Otherwise, we have s=λm+1
Xm i=1
λi
λm+1
si+λm+1sm+1
and λi
λm+1 ≥0.
Thus, by induction, the linear combination s′=
Xm i=1
λi
λm+1
si
belongs toC. SinceC is a cone and containsS, it follows that the elements =λm+1s′+λm+1sm+1
belongs toC, as required. ⊓⊔