An update on the Hirsch conjecture: Fifty-two years later

The Hirsch conjecture was posed in 1957 in a letter from Warren M. Hirsch to George Dantzig. It states that the graph of a d-dimensional polytope with n facets cannot have diameter greater than n− d. That is to say, we can go from any vertex to any other vertex using at most n− d edges. Despite being one of the most fundamental, basic and old problems in polytope theory, what we know is quite scarce. Most notably, no polynomial upper bound is known for the diameters of polytopes. In contrast, very few polytopes are known where the bound n − d is attained. This paper collects several results and remarks both on the positive and on the negative side of the conjecture. Some proofs are included, but only those that we hope are accessible to a general mathematical audience without introducing too many technicalities.