**Conjecture**The average diameter of a bounded cell of a simple arrangement defined by hyperplanes in dimension is not greater than .

Let be a simple arrangement formed by hyperplanes in dimension . The number of bounded cells of is . Let denote the average diameter of a bounded cell of ; that is, Let denote the largest possible average diameter of a bounded cell of a simple arrangement defined by inequalities in dimension .

We have [DTZ,DX]:

If the conjecture of Hirsch holds, then .

for .

for .

for .

## Bibliography

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

[DX] A. Deza and F. Xie: Hyperplane arrangements with large average diameter. Centre de Recherches Mathematiques and American Mathematical Society series (to appear).

* indicates original appearance(s) of problem.