The Alon--Milman Theorem for non-symmetric bodies

A classical theorem of Alon and Milman states that any $d$ dimensional centrally symmetric convex body has a projection of dimension $m\geq e^{c\sqrt{\ln{d}}}$whichiseitherclosetothe$m$-dimensional Euclidean ball or to the $m$-dimensional cross-polytope. We extended this result to non-symmetric convex bodies.