TITLE: Sum of strictly convex and convex functions
QUESTION [13 upvotes]: My prof mentioned that the sum of strictly convex and convex functions is strictly convex, Im having trouble swallowing that, is it accurate?

REPLY [11 votes]: Assume $f$ is convex and $g$ is stricly convex. Let $0<\theta < 1$. Then calculate that
$$
\begin{align}
   (f+g)(\theta x + (1-\theta) y) &= f(\theta x + (1-\theta) y) + g(\theta x + (1-\theta) y)  \\
  &<  \theta f(x) + (1-\theta) f(y) + \theta g(x) + (1-\theta) g(y) \\
    &=  \theta (f+g)(x) + (1-\theta) (f+g)(y)
\end{align}
$$
where there is strict inequality because the inequality is strict in one case (and not necessarily strict in the other case).