The number of extreme points of tropical polyhedra

  • Authors:

  • Xavier Allamigeon;Stéphane Gaubert;Ricardo D. Katz

  • Affiliations:

  • Direction du Budget, 4ème sous-direction, Bureau des transports, 139 rue de Bercy, Télédoc 278, 75572 Paris Cedex 12, France;INRIA and CMAP, ícole Polytechnique, 91128 Palaiseau Cedex, France;CONICET, Instituto de Matemática "Beppo Levi", Universidad Nacional de Rosario, Avenida Pellegrini 250, 2000 Rosario, Argentina

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2011

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Abstract

The celebrated upper bound theorem of McMullen determines the maximal number of extreme points of a polyhedron in terms of its dimension and the number of constraints which define it, showing that the maximum is attained by the polar of the cyclic polytope. We show that the same bound is valid in the tropical setting, up to a trivial modification. Then, we study the tropical analogues of the polars of a family of cyclic polytopes equipped with a sign pattern. We construct bijections between the extreme points of these polars and lattice paths depending on the sign pattern, from which we deduce explicit bounds for the number of extreme points, showing in particular that the upper bound is asymptotically tight as the dimension tends to infinity, keeping the number of constraints fixed. When transposed to the classical case, the previous constructions yield some lattice path generalizations of Gale's evenness criterion.