Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be strictly convex and differentiable. Is $f$ strongly convex when restricted to a closed and bounded interval $[a,b]$?
This is true if $f$ is smooth but am wondering if it still holds when $f$ doesn't have a second derivative.
$f(x)=x^4$ is strictly convex, but not strongly convex on any interval including $0$ since its second derivative $12x^2$ won't be bounded above some positive constant near $0.$ [Note this example appeared on the wiki page you cited.]