Can convex function contains fraction term? I have one specific function. Can someone help me clarify whether it is a convex function or not?

Question

I hope someone could help me clarify my question.

Is this function a convex function? $(\frac{w_1 + w_2}{w_1+w_2+w_3+w_4} - \frac{1}{2})^2$

$w_1, w_2, w_3, w_4$ are variables.

Basically, multi-variables make me hard to draw a conclusion.

I appreciate any help!

Answer 1

This is neither convex nor concave.

For instance, consider the special case $$(\frac{m_1}{m_1+m_2}-\frac{1}{2})^2$$ Its Hessian evaluated at $m_1 = 1, m_2 = 2$ has one negative eigenvalue and one positive eigenvalue. Hence the Hessian is indefinite there. Hence the function is neither convex nor concave.