How a hyperplane separates a convex set. Please give me mathematical and geometrical representation and also I want to know the consequences after separations?
Actually, I was reading a scholarly paper, there I found these lines: $v \cdot (y − x) \le 0, \ \forall y \in C$ (a convex set subset of $\Bbb R^n$), which implies that $C$ is separated from $x+v$ by the hyperplane $H = \{ z \in \Bbb R^n : v \cdot (z − x) = 0 \}$ passing through $x$ and orthogonal to $v$.
I didn't get meaning of these lines. Please give the explanation of these lines with geometrically representations.
EDIT: Here's a picture of the situation in the "scholarly paper". The convex set $C$ is in red, and the hyperplane $H$ is blue. The vector $v$ is $(1,0)$, so $H$ is a vertical line passing through $x$. The inequality $v \cdot (y - x) \le 0$ for $y \in C$ says that $C$ is to the left of this hyperplane (possibly with some points on the hyperplane). $x+v$ is to the right of the hyperplane.