I'm working through a support vector machines tutorial. In eventually deriving the solvable objective function, the following objective function (to be maximized) was proposed, but dismissed as non-convex (i.e., it's negative is non-convex). Why is it non-convex?
$$ \newcommand{\norm}[1]{\lVert#1\rVert} f(\hat{\gamma}, w, b) = \frac{\hat{\gamma}}{\norm{w}} = \min_{i=1, \dots, m} y^{(i)} \left( \frac{w^{T}}{\norm{w}} x^{(i)} + \frac{b}{\norm{w}} \right) $$
where the label variable $y^{(i)} \in \{-1,1\}$, the feature vector $x^{(i)} \in \mathbb{R}^{n}$, $w \in \mathbb{R}^{n}$, and $b \in \mathbb{R}$.
The (convex) constraints are: $$ y^{(i)} (w^{T}x^{(i)} + b) \ge \hat{\gamma},\; i=1, \dots, m. $$
As noted in the comments (thanks Michael Grant), I was overcomplicating things. It's easy to show non-convexity with respect to $w$. First, consider $w \in \mathbb{R}^{1}$ and show that (the negative of) $f$ is not convex by checking a couple of values of $w$. Secondly, note that $f$ is undefined for $w = \vec{0}$, implying that the $w$ domain of $f$ is non-convex in $\mathbb{R}^{n}$.