A positive semidefinite approximation of the symmetric traveling salesman polytope

For a convex body B in a vector space V, we construct its approximation P-k, k = 1, 2,..., using an intersection of a cone of positive semidefinite quadratic forms with an affine subspace. We show that Pk is contained in B for each k. When B is the Symmetric Traveling Salesman Polytope on n cities T-n we show that the scaling of Pk by n/k + O(1/n) contains T, for k <= [n/2]. Membership for Pk is computable in time polynomial in n (of degree linear in k). We also discuss facets of Tn that lie on the boundary of Pk and we use eigenvalues to evaluate our bounds.