(quasi) convexity x/y

Question

Hej,

I have the function $\frac{x}{y}$ on the domain $R_{++}$. The Hessian matrix is - as I have calculated it - positive semidefinite. But I'm not really sure, if the function is really convex at all on the domain.

Thanks for any help.

Answer 1

Something went wrong with your calculation, because the Hessian matrix $$ \begin{pmatrix} 0 & -1/y^2 \\ -1/y^2 & 2/y^3 \end{pmatrix} $$ has negative determinant.

There is a similar function with positive semidefinite Hessian in the positive quadrant, namely $v(x,y)=x^2/y$. The Hessian is $$ \begin{pmatrix} 2/y & -2x/y^2 \\ -2x/y^2 & 2x^2/y^3 \end{pmatrix} = \frac{2}{y^3}\langle x,y \rangle \otimes \langle x,y \rangle \ge 0 $$ Given its simple form, I wonder if there is any "obvious" reason for the convexity of $v$.