Isomorphic random subspaces and quotients of convex and quasi-convex bodies

We extend the results of [LMT] to the non-symmetric and quasi-convex cases. Namely, we consider a finite-dimensional space endowed with the gauge of either a closed convex body (not necessarily symmetric) or a closed symmetric quasi-convex body. We show that if a generic subspace of some fixed proportional dimension of one such space is isomorphic: to a generic quotient of some proportional dimension of another space then for any proportion arbitrarily close to 1, the first space has a lot of Euclidean subspaces and the second space has a lot of Euclidean quotients.