Maximum likelihood estimation in $d$-dimensional Euclidean space of a ball I have a set of points in $d$-dimension Euclidean space drawn from a ball centered at point $c$ and with radius $r$ which are unknown. I want help in formulating the maximum likelihood estimator of $r$ and $c$.

If they're uniformly distributed in the ball, then the value of the probability density at every point in the ball is equal to the reciprocal of the volume of the ball. The values of $c$ and $r$ that maximize that density, subject to the constraint that all of the observed points lie within the ball, are the values of $c$ and $r$ that give you the smallest possible ball containing the observed points.
That reduces the problem to one of geometry. Possibly more that is of interest could be said about the geometry problem.
