Integral geometry of higher-dimensional polytopes and the average case in combinatorial optimization

We consider the average case behavior of a linear optimization problem on various series of combinatorially interesting polytopes. From general results of integral geometry it follows that for all but an asymptotically negligible fraction of linear functions a polytope can be replaced by a pair of concentric balls with asymptotically equal radii so that the optimal value of the linear function on the polytope is in the interval between the optimal values of the linear function on these balls. In particular, we show that the average case behavior of the assignment problem, traveling salesman problem, and, generally speaking, of any optimization problem on a polynomial fraction of all permutations is the same