Margin for a linear separator

Suppose we have a set of linearly separable points $X = X_1 \cup X_2 \subset \mathbb{R}^n$ such that $w \in \mathbb{R}^n$ and $b \in \mathbb{R}$ separate them. We define the margin of the hyperplane $H = (w, b)$ with respect to $X$ to be

$$\rho = \min_{x \in X} \left \vert \frac{(x - bw) \cdot w}{\vert \vert w \vert \vert} \right \vert = \min_{x \in X} \left \vert \frac{x \cdot w - \vert \vert w \vert \vert^2 b}{\vert \vert w \vert \vert} \right \vert.$$

In other words this is the smallest distance from a point in $X$ to the hyperplane $H$.