Efficient algorithm to compute the diameter of a convex set?
I suspect that if the points in your set are expressed as $Ax \le b$ and you're considering the Euclidean distance, then you can solve the following quadratic program with the ellipsoid method:
$$
\max  \sum_i (x^2_i+y^2_i-2x_i y_i) \\
Ax \le b \\
Ay \le b.
$$
Where we used the fact that optimizing $|x-y|$ is the same as optimizing $ \frac{1}{2} |x-y|^2 = \frac{1}{2} \sum_i (x_i-y_i)^2 = \frac{1}{2} \sum_i (x_i^2+y_i^2 - 2x_iy_i)$.
The objective function can also be written as $\frac{1}{2} z^T Q z$, where $z=(x,y)$ and:
$$
Q= \begin{bmatrix}
I &amp; -I \\
-I &amp; I
\end{bmatrix}.
$$