Orthogonal Projection of a convex hypersurface

Question

Suppose I have an $n-1$ dimensional convex hypersurface $\Omega$ in $\mathbb{R}^n$ and I know $o\notin\Omega$ and $\Omega$ is contained in some closed half space.

Is it true that I can always choose a direction $v$ such that

1. $\Omega \subset \{x\in \mathbb{R}^n: x\cdot v\leq 0\}$ and
2. $o$ is in the the interior of the image of the orthogonal projection of $\Omega$ onto $v^\perp$?