I need to understand if $$\min_{x,y}\left\{\sqrt{1+x^2}-\sqrt{1-y^2}\,\middle|\,\frac{\sqrt{(x-1)^{2}+y^2}}{\sqrt{x^2+(y-2)^2}}\leq\frac12,\ |y|\leq1\right\}$$is convex problem. I found that the domain$$C=\left\{ \begin{pmatrix}x\\y\end{pmatrix}\in\mathbb{R}^{2}\,\middle|\,\frac{\sqrt{(x-1)^{2}+y^{2}}}{\sqrt{\sqrt{x^{2}+(y-2)^{2}}}}\leq\mu\right\} \cap\left\{ \begin{pmatrix}x\\y\end{pmatrix}\in\mathbb{R}^{2}\,\middle|\,|y|\le1\right\}$$ is convex and I need to figure out if $f\left(\begin{pmatrix}x\\y\end{pmatrix}\right)=\sqrt{1+x^{2}}-\sqrt{1-y^{2}}$ is a convex function in that domain.
I found the gradient of $f$ is: $$\nabla f\left(\left(\begin{array}{c} x\\ y \end{array}\right)\right)=\left(\begin{array}{c} \frac{x}{\sqrt{1+x^{2}}}\\ \frac{y}{\sqrt{1-y^{2}}} \end{array}\right)$$ and the hessian of f is $$\nabla^{2}f\left(\left(\begin{array}{c} x\\ y \end{array}\right)\right)=\left(\begin{array}{cc} \frac{1}{\sqrt{1+x^{2}}}-\frac{x^{2}}{\left(\sqrt{1+x^{2}}\right)^{3}} & 0\\ 0 & \frac{1}{\sqrt{1-y^{2}}}+\frac{y^{2}}{\left(\sqrt{1-y^{2}}\right)^{3}} \end{array}\right)$$
I want to show that the hessian is not psd and therefor $f$ is not convex,but I don't rellay sure how to do that..
Thanks!!!