I have a set of points in $\mathbb{R}^2$ and I need to find the smallest enclosing circle, i.e. the circle with the smallest radius that contains all of the points belonging to the set.
I have the following solution but I cannot prove its correctness or wrongness, I suspect it is wrong.
I find four points in the set:
$L=(x_L,y_L)$ has the smallest $x$ coordinate value,
$R=(x_R,y_R)$ has the largest $x$ coordinate value,
$B=(x_B,y_B)$ has the smallest $y$ coordinate value,
$T=(x_T,y_T)$ has the largest $y$ coordinate value.
$L$, $R$, $B$, $T$ stand for Left, Right, Bottom, Top.
Then for each of the four points I compute the distance to the other three points, for example I compute the distance from $L$ to $R$ like this
$d(L,R)=\sqrt{(x_L-x_R)^2+(y_L-y_R)^2}$
Now I have all the six distances: $d(L,R)$, $d(L,B)$, $d(L,T)$, $d(R,B)$, $d(R,T)$, $d(B,T)$.
I find the maximum distance of the six distances, let's call it $d(X,Y)$, and I say that the radius of the minimum enclosing circle is $\frac{d(X,Y)}{2}$ and its center is the midpoint of the segment $\overline{XY}$.
Is it a wrong solution? Why?