Suppose we have a univariate function $f(t), t\in [0,1]$. We define \begin{align} G(p) = \sup_{t\in [0,1]} p f(t) + (1-p) f(1-t), \text{ for } p\in [0,1]. \end{align} Clearly $G(p)$ is a convex function in $p$ since it is the supremum of a family of convex functions in $p$. However, we want a stronger requirement: we would like to make sure that $G(p)$ is strictly convex at point $p = 1/2$. In other words, we want to guarantee that for any $x,y\in [0,1], x \neq 1/2, y\neq 1/2$, any $u\in (0,1)$ such that $u x + (1-u) y = 1/2$, we have \begin{align} G(1/2) < u G(x) + (1-u) G(y). \end{align}
The question is, what is the most general condition on $f$ such that this requirement is satisfied?
It is easy to verify that for special examples such that $f(t) = \log t$ or $f(t) = t$, it is true.
Conjecture: it is true whenever $f$ is a concave function and $f'(1/2) \neq 0$.