Quasi-Euclidean classification of alcoved convex polyhedra

ABSTRACT

We give the quasi-Euclidean classification of the maximal (with respect to the f-vector) alcoved polyhedra. The f-vector of these maximal convex bodies is $(20,30,12)$, so they are simple dodecahedra. We find eight quasi-Euclidean classes. This classification, which preserves angles, is finer than the known combinatorial classification (found in 2012 by Jiménez and de la Puente), which has only six classes. Each alcoved polyhedron $\mathcal{P}$ is represented by a unique visualized idempotent matrix A. Some 2-minors of A are invariants of $\mathcal{P}$: they are the tropical edge-lengths of $\mathcal{P}$.

COMMUNICATED BY:

• Stéphane Gaubert

Keywords:

• Tropical algebra
• max-plus algebra
• normal matrix
• idempotent matrix
• visualized matrix
• alcoved polyhedron
• dodecahedron
• convex body
• quasi-Euclidean class
• perturbation
• invariant
• edge-length
• tropical distance

AMS Classifications:

• 52B10
• 52B12
• 15A80

Acknowledgments

I am deeply grateful to the referee for careful reading and interest. His/her suggestions and patience have been a great help to bring this paper to light. I also thank my friend P.L. Clavería for producing 3-D models of many alcoved dodecahedra and for checking many computations.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

M. J. de la Puente  http://orcid.org/0000-0002-3549-0973

Notes

1 We will use the alien element $-\mathrm{\infty }$ as little as possible.

2 $\mathrm{Instead}\mathrm{of}\mathrm{t}-\mathrm{conv}\left(A\right),\mathrm{we}\mathrm{could}\mathrm{use}\mathrm{as}\mathrm{well}\mathrm{t}-\mathrm{cone}\left(A\right),\mathrm{the}\mathrm{definition}\mathrm{being}\mathrm{analogous}\mathrm{to}\mathrm{t}-\mathrm{conv}\left(A\right),\mathrm{except}\mathrm{that}{\lambda }_j\mathrm{runs}\mathrm{in}\mathbb{R}\cup \{-\mathrm{\infty }\}.\mathrm{The}\mathrm{set}\mathrm{t}-\mathrm{cone}\left(A\right)\mathrm{is}\mathrm{a}\mathrm{tropical}\mathrm{cone}\mathrm{and}\mathrm{the}\mathrm{difference}\mathrm{between}\mathrm{both}\mathrm{sets}\mathrm{is}\mathrm{just}\mathrm{the}\mathrm{point}(-\mathrm{\infty },\dots ,\mathrm{\infty }{)}^T.\mathrm{We}\mathrm{get}\mathcal{P}\left(A\right)=\{x\in {\mathbb{R}}^n:x_n=0\}\cap \mathrm{t}-\mathrm{cone}\left(A\right).$.

3 Indeed, for each $i\in \left[n\right]$, take ${\lambda }_i=0$, ${\lambda }_j<0$ and use $a_{ii}=0$ and $a_{ij}\le 0$, to obtain $x_i=\underset{k\in \left[n\right]}{max}\{{\lambda }_k+a_{ik}\}=0$.

4 It implies that, for a NI matrix, vertex labels in the North Cask of $\mathcal{P}$ follow the same cyclic sequence for every polyhedron, when going around $\mathcal{N}$. This will be used in p. 23.

5 The order of digits is unimportant.

6 Recall that $\underset{\_}{123}=\mathcal{N}$ and $\underset{\_}{4}=\mathcal{S}$, i.e., the South Pole is a generator in $\mathcal{P}$.

7 The order of digits is important.

8 If n>4, we do not have a general rule to label the non-principal vertices of $\mathcal{P}$.

9 This is related to, but different from, the tropical determinant (also called tropical permanent).

10 In this section we take n=4, because we do not know how to make definition 5.2 in more generality.

11 This makes sense, because a box is an unperturbed alcoved polyhedron.

12 ${\mathcal{G}}_3$ is isomorphic to a dihedral group of order 12.

13 A chiral copy of a rubber glove G is obtained by turning G inside out. Alternatively, we cut G along a meridian, we fold the two pieces inside out and then we glue them again. A chiral copy of the cube $\mathcal{Q}$ is obtained similarly.

14 This is not true for all $g\in {\mathcal{G}}_4$; for instance, take $g=+\left(1234\right)$.

15 The action of a group G on a set S is a map $G\times S\to S$, $(g,s)\mapsto g\cdot s$, such that (a) $\mathrm{id}\cdot s=s$, all $s\in S$ and (b) $g_1\cdot (g_2\cdot s)=\left(g_1{g}_2\right)\cdot s$, all $g_1,g_2\in G$, all $s\in S$.

16 $\mathrm{d}(p,q)$ is the maximum of the Chebyshev distance ${\mathrm{d}}_{Ch}(p,q):=\underset{i,j\in \left[n\right]}{max}\{|p_i-q_i|\}$ and a tropical version of the Hilbert projective distance $\mathrm{d}(p,q{)}_{pr.H}:=\underset{i,j\in \left[n\right]}{max}\{|p_i-q_i-p_j+q_j|\}$.

17 We explain the instance i=2, the other ones being similar (see Figure 7, right). The coordinates of the generator $\underset{\_}{3}$ are $\left(\begin{array}{c}{a}_{13}\\ a_{23}\\ 0\end{array}\right)$ and ${\delta }_3+{\rho }_3=-a_{13}$ follows from (29(29) ${\delta }_{i+1}+{\rho }_{i+1}=\left\{\begin{array}{l}-a_{i-1,i+1},\mathrm{if}\mathrm{case}1\mathrm{above},\\ -\mathrm{\infty },\mathrm{otherwise},\end{array}\right.{ϵ}_{i-1}+{\lambda }_{i-1}=\left\{\begin{array}{l}-\mathrm{\infty },\mathrm{if}\mathrm{case}1\mathrm{above},\\ -a_{i+1,i-1},\mathrm{otherwise},\end{array}\right.$(29) ), ${ϵ}_3+{\lambda }_3=-a_{23}$ follows from (31(31) $\begin{array}{rl}{ϵ}_{i+1}+{\lambda }_{i+1}& =\left\{\begin{array}{l}-a_{i,i+1}\\ -\mathrm{\infty }.\end{array}\right.\end{array}$(31) ). Besides, ${\rho }_3+|a_{21}|=|a_{23}|$ follows from (27(27) $a_{i,4;i-1,i+1}=a_{i,i-1}-a_{i,i+1},i\in \left[3\right].$(27) ) and case (1)a. Similarly, we have ${\lambda }_3+|a_{12}|=|a_{13}|$.

18 Note 1=10−12+3, i.e., a North Cask has the Euler characteristic of a closed disc.

19 Notice the signs in (42(42) $\begin{array}{rl}& -\mathrm{id}\mapsto +\left(14\right)\left(25\right)\left(36\right)\end{array}$(42) )–(46(46) $-\left(23\right)\mapsto -\left(16\right)\left(25\right)\left(34\right)$(46) )!

20 We say that $\underset{\_}{4i},\underset{\_}{i4}$ gives one inversion, and $\underset{\_}{i4},\underset{\_}{4i}$ gives no inversion.

21 By remark 9.21, the Equatorial Belt is not to be taken int account.

22 The h-vector considered here has nothing to do with the h-vector found in the literature on f-vectors.

Additional information

Funding

Partially supported by Ministerio de Economía y Competitividad, Proyecto I+D MTM2016-76808-P and by UCM research group 910444.