On isometric immersion of closed manifolds of nonnegative curvature

Let $M^n$ be a closed manifold. Assume that an immersion $f\colon M^n\to\mathbb R^N$ induces a $C^2$-smooth metric of nonnegative curvature or a polyhedral metric of nonnegative curvature on $M^n$. If this nonnegativness is left invariant under every affine transformation of $\mathbb R^N$, then $f$ is an embedding on the boundary of a $C^2$-smooth convex body (a convex polyhedron correspondingly) in some $\mathbb R^{n+1}\subset\mathbb R^N$.