Convex function definition

Question

This just occurred to me but I can't figure out which step is wrong. It is well-known that the convex function's definition is $\alpha f(x_1) + (1-\alpha) f(x_2) \ge f(\alpha x_1 + (1-\alpha) x_2)$. But if we just look at the geometric interpretation (e.g. https://en.wikipedia.org/wiki/File:ConvexFunction.svg), we can see that the segment $\alpha f(x_1) + (1-\alpha) f(x_2)$ is always above $f(x)$ in the range $[x_1, x_2]$. Therefore, my question is why the convex function does not have this stronger property $\alpha f(x_1) + (1-\alpha) f(x_2) \ge f(\beta x_1 + (1-\beta) x_2)$ for all $\alpha, \beta$ which are not necessarily the same? Can anyone give me a convex function example that does not satisfy the stronger constraint I just wrote?

Answer 1

Take for instance $$f(x)=x^2,\quad x_1=0,\quad x_2=1,\quad \alpha=1,\quad\beta=0. $$