The sup is as following:
$c_f = sup_{x,s\in D} \ f(y) - f(x) - (y-x)^Tb$
where $y=x+\alpha(s-x)$, $\alpha \in (0,1 )$ is constant and $b$ is a constant vector. $D$ is a convex compact set and $f:R^n\rightarrow R$ is a convex function. Is $c_f$ bounded? I think it is because everything in the above equation is bounded ($y$ is on the line segment between $x$ and $s$.). Please help me to prove or disprove that $c_f$ is bounded. Thanks:)