Recently I started playing with n-spheres for my classification problem and end up with some weird results.
Let's say that we have m points $p_i$, each is n-dimensional. I want to find the minimum covering n-sphere, centered at $C_c$ and has radius $r_c$. And it can be found easily by solving following equation:
$${p_i},{C_c} \in {R^n},i = 1 \ldots m$$
$$\mathop {\min }\limits_{{r_c} \ge {{\left\| {{p_{_i}} - {C_c}} \right\|}_2}} {}_{\left( {{r_c},{C_c}} \right)}\left\{ {{r_c}} \right\} $$
This is an outer n-sphere which covers all $p_i$. The solution is unique and exists.
Now I also want to find a maximum n-sphere which doesn't have any $p_i$ in it but centered at $C_f$ which is inside the covering sphere. I call this as a maximum fitting n-sphere, and I can solve it using following equation:
$$\mathop {\max }\limits_{\begin{array}{*{20}{c}}{{r_f} \le {{\left\| {{p_{_i}} - {C_f}} \right\|}_2}}\\{{r_c} \ge {{\left\| {{C_c} - {C_f}} \right\|}_2}}\end{array}} {}_{\left( {{r_f},{C_f}} \right)}\left\{ {{r_f}} \right\} $$
This one is not as easy as above to solve since it is not positive semi definite but I can find some solutions.
In summary, I want to find n-spheres shown like below image:
Blue: Minimum Covering n-sphere
Orange: Maximum Fitting n-sphere
Now the problem, if all my equations are correct, I always end up with something $r_f$ > $r_c$. To me this is not logical. Could you please briefly verify my equations and tell me why always end up with n-spheres which have bigger radius inside than outside?
The maximum fitting $n$-sphere is centered inside the minimum fitting $n$-sphere. However, that doesn't mean it needs to be inside the minimum n-sphere. It's radius can be up to twice as big as $r_c$ . The simplest example illustrating the problem occurs with $n=2$ and $3$ points:
A more natural definition for the "maximum fitting n-sphere" could be "the largest sphere which fits inside the convex cover (convex hull) of our points". This has a unique solution and can be solved analytically, just like the minimum fitting $n$-sphere.