Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  introduce the property of k-decomposability for simplicial complexes which, if satisfied by the dual simplicial complex of a convex polyhedron, implies that the diameter of the polyhedron is bounded by a polynomial of degree k + 1 in the number of facets. These properties form a hierarchy, each one implying the next. The strongest, vertex decomposability, implies the Hirsch conjecture; the akest is equivalent to shellability.  show that several cases in which the Hirsch conjecture has been verified can be handled by these methods, which also give the shellability of a number of simplicial complexes of combinatorial interest.  conclude with a strengthened form of the property of shellability which would imply the Hirsch conjecture for polytopes.
['J. Scott Provan', 'Louis J. Billera']
I am writing a research paper on a idea Decompositions of Simplicial Complexes Related to Diameters of Convex Polyhedra
[{"from": "human", "value": "I am writing a research paper on a idea Decompositions of Simplicial Complexes Related to Diameters of Convex Polyhedra"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  introduce the property of k-decomposability for simplicial complexes which, if satisfied by the dual simplicial complex of a convex polyhedron, implies that the diameter of the polyhedron is bounded by a polynomial of degree k + 1 in the number of facets. These properties form a hierarchy, each one implying the next. The strongest, vertex decomposability, implies the Hirsch conjecture; the akest is equivalent to shellability.  show that several cases in which the Hirsch conjecture has been verified can be handled by these methods, which also give the shellability of a number of simplicial complexes of combinatorial interest.  conclude with a strengthened form of the property of shellability which would imply the Hirsch conjecture for polytopes."}]