Convex hypersurface

Question

Let $(M,\lambda)$ be a contact manifold with contact form $\lambda$. A convex hypersurface is defined to be a hypersurface $\Sigma$ with a vector field $v$ on $M$ which is transverse to $\Sigma$ and preserves $\lambda$, that is, the Lie derivative $L_v\lambda =0$. My question is what does the name convex have to do with the usual definition of a convex subset of $\mathbb{R}^n$?