When do we have equality in Jensen's inequality?

Question

I'm talking about the finite form of the inequality:

$$f(q_1x_1+q_2x_2+\cdots+q_nx_n)\leq q_1f(x_1)+q_2f(x_2)+\cdots+q_nf(x_n)$$ with $$\sum_{i=1}^{n}{q_i}=1, q_i\geq 0$$ (Obviously the form for convex functions.)

I'm just wondering when does the equality happen.

Answer 1

One way equality can occur is if $x_1 = \cdots = x_n$. Another way the equality can occur is if $f$ is linear.