Separation properties of convex set in the n-dimensional Euclidean space

Question

Let S be a nonempty closed convex set in Rn, not containing the origin. I would like to find a hyperplane that strictly separates S and the origin. Does it exists?

Answer 1

Hint

Name $x$ the orthogonal projection of $x$ onto $S$. By hypothesis $x$ exists as $S$ is a non empty closed convex of the finite dimensional space $\mathbb R^n$. Also $x \neq 0$.

The mid hyperplane $H$ of $[0, x]$ segment is one hyperplane answering your question.