How can the Rosenbrock function be non-convex?

Question

I am dealing with the Rosenbrock function

$$ f(x,y) = (1-x)^2 + 100(y-x^2)^2$$

I read online that this is a non convex function. However when I watch its graph, it looks like a convex function. Can someone explain me what I am doing wrong?

Answer 1

For example $$f\left(\frac{(-1,1)+(0,0)}2\right)=\frac{17}2>\frac52=\frac{f(-1,1)+f(0,0)}2$$ so $f$ is not convex. More systematically, one could note that the Hessian of $f$ is not always positive semi-definite.