I came across these 2 seemingly contradictory statements in the book 'Convex Optimization' by Boyd.
and
So the top image is said to be concave. For the simplest case where $n=1$ the function is the same as described below as a convex function. So what am I missing here?
In the top equation (with $n = 1$), the 'inner' function is $$f(w,x) = w \, (a^\top x - b)^2.$$ This function is not convex. However, it is linear (thus concave) in the variable $w$. Thus, $$g(w) = \inf_x f(w,x)$$ is the infimum of the family of convex functions $\{f(\cdot, x) \mid x\}$.
In the bottom equation, $f$ is assumed to be convex (i.e., convex in $(x,y)$). This results in $g$ being convex.