# Last Point of Exit from the 2-D Positive Quadrant

I define the functions $f_i(\mathbf{u}),i=1,2$ and $g_i(\mathbf{u}),i=1,2$ where $\mathbf{u}\in\mathbb{C}^{N}$ is the unit-norm vector. Thus, this functions are defined over the unit norm $N-$sphere. Now all of them are positive over the unit-norm shpere. Consider the set \begin{align} \mathbb{S}(t)=\{[f_1(\mathbf{u})-tf_2(\mathbf{u}),g_1(\mathbf{u})-tg_2(\mathbf{u})]\in \mathbb{R}^2,\mathbf{u}^H\mathbf{u}=1\} \end{align} Now $\mathbb{S}(t)$ is a convex, continuous and closed set (solid body) regardless of $t$(this is given and comes from the nature of those functions, Read below for more details). Note that at $t=0$, $\mathbb{S}(0)$ is totally in positive quadrant. As one can see, as $t$ increases, $\mathbb{S}(t)$ gradually moves away from the positive quadrant, and after some particular value of $t$, it totally exits positive quadrant. I am interested in this particular $t$. Can we make general comment on it? Is there any method to compute it.

MORE DETAILS Please read this only if you want to know more about $f_i$ and $g_i$. Let $A_i$ and $B_i$ be positive-definite matrices for $i=1,2$. Then $f_i(\mathbf{u})=\mathbf{u}^HA_i\mathbf{u}$ and $g_i(\mathbf{u})=\mathbf{u}^HB_i\mathbf{u}$. The fact that $\mathbb{S}(t)$ is convex regardless of $t$ then naturally follows from Toeplitz-Hausdorff Theorem.

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