  For a polytope P, the Chvatal closure P' is obtained by simultaneously
strengthening all feasible inequalities cx <= b (with integral c) to cx <=
floor(b). The number of iterations of this procedure that are needed until the
integral hull of P is reached is called the Chvatal rank. If P is a subset of
[0,1]^n, then it is known that O(n^2 log n) iterations always suffice
(Eisenbrand and Schulz (1999)) and at least (1+1/e-o(1))n iterations are
sometimes needed (Pokutta and Stauffer (2011)), leaving a huge gap between
lower and upper bounds.
  We prove that there is a polytope contained in the 0/1 cube that has Chvatal
rank Omega(n^2), closing the gap up to a logarithmic factor. In fact, even a
superlinear lower bound was mentioned as an open problem by several authors.
Our choice of P is the convex hull of a semi-random Knapsack polytope and a
single fractional vertex. The main technical ingredient is linking the Chvatal
rank to simultaneous Diophantine approximations w.r.t. the L1-norm of the
normal vector defining P.
