Since it is known that you can sort $n$ numbers by solving a certain kind of linear program - doesn't this imply a lower bound on the complexity of solving linear programs in general via the lower bound for sorting? But it's a bit tricky since the linear program in question has $n^2$ variables and $2n$ equalities. There is another method I know of that has $n$ variables and $2^n$ inequalities (although that number of inequalities can surely be reduced) - but I don't know how you could translate this into a lower bound...
For the linear program with $n^2$ variables and $2n$ equalities see this article (right at the bottom). If you are skeptical that this formulation does indeed sort the numbers, here is a matlab implementation.
Actually this particular formulation can be made more succint. Consider the optimization problem
max sum(D*P*x) over all permutation matrices P
here D is the matrix with 1,2,...n on the diagonal. Any increasing sequence would work as well - but 1,2,...n is pretty clear. x = (x1,x2,...,xn) is the vector of values to be sorted. So after applying some permutation you weight the value with D. So we want to solve this optimization problem for all the values of P (since D and x are known). sum() is the sum over all vector components. Since all the equations for this turn out to be linear this problem is solvable as a linear program.