-
A function is convex if
$f(\theta x + (1-\theta) y) \leq \theta f(x) + (1-\theta) f(y)$,
$\theta \in [0,1]$ -
A function is strictly convex if
$f(\theta x + (1-\theta) y) < \theta f(x) + (1-\theta) f(y)$, $\theta
\in (0,1)$
What is the big deal with definition of $\theta$ between these two definitions?
Why must $\theta$ not take on the boundary values of $0$ and $1$ in the definition of strict convexity?
Best Answer
The $[0,1]$ or $(0,1)$ is not that important. You can write $(0,1)$ instead of $[0,1]$ in the first definition.
What is important is that the $\leq$ sign changes to $<$.
Of course, the $<$ inequality cannot hold for $\theta=0$, since the inequality then becomes $f(y)<f(y)$, which is not true for any function, so you cannot write $[0,1]$ instead of $(0,1)$ in the second definition.