Importance: Medium ✭✭
 Author(s): Deza, Antoine Terlaky, Tamas Zinchenko, Yuriy
 Subject: Geometry » Polytopes
 Keywords: curvature polytope
 Posted by: deza on: September 30th, 2007
Conjecture   The order of the largest total curvature of the primal central path over all polytopes defined by inequalities in dimension is .

Let denote the total curvature of the central path corresponding to the linear optimization problem . The quantity can be regarded as the continuous analogue of the edge-length of the shortest path between a pair of vertices. Considering the largest over all possible we obtain the quantity , referred to as the curvature of a polytope. Following the analogy with the diameter, let be the largest total curvature of the primal central path over all polytopes defined by inequalities in dimension .

Holt and Klee~[HK] showed that, for , the conjecture of Hirsch is tight. We have the following continuous analogue of the result of Holt and Klee:

[DTZa] , that is, is bounded below by a constant times .

The special case of of the conjecture of Hirsch is known as the -step conjecture, and it has been shown by Klee and Walkup~[KW] that the -step conjecture is equivalent to the Hirsch conjecture. We have the following continuous analogue of the result of Klee and Walkup:

[DTZb] If the order of the curvature is less than the dimension for all polytope defined by inequalities and for all , then the order of the curvature is less that the number of inequalities for all polytopes; that is, if for all , then .

## Bibliography

[DMS] J.-P. Dedieu, G. Malajovich and M. Shub: On the curvature of the central path of linear programming

*[DTZa] A. Deza, T. Terlaky and Y. Zinchenko: Polytopes and arrangements : diameter and curvature. Operations Research Letters (to appear).

[DTZb] A. Deza, T. Terlaky and Y. Zinchenko: The continuous d-step conjecture for polytopes. AdvOL-Report 2007/16, McMaster University (2007).

[HK] F. Holt and V. Klee: Many polytopes meeting the conjectured Hirsch bound. Discrete and Computational Geometry 20 (1998) 1--17.

[KW] V. Klee and D. Walkup: The -step conjecture for polyhedra of dimension . Acta Mathematica 133 (1967) 53--78.

* indicates original appearance(s) of problem.