  In this paper we analyze the extension of the classical smallest enclosing
disk problem to the case of the location of a polyellipsoid to fully cover a
set of demand points in $\mathbb{R}^d$. We prove that the problem is
polynomially solvable in fixed dimension and analyze mathematical programming
formulations for it. We also consider some geometric approaches for the problem
in case the foci of the polyellipsoids are known. Extensions of the classical
algorithm by Elzinga-Hearn are also derived for this new problem. Moreover, we
address several extensions of the problem, as the case where the foci of the
enclosing polyellipsoid are not given and have to be determined among a
potential set of points or the induced covering problems when instead of
polyellipsoids, one uses ordered median polyellipsoids. For these problems we
also present Mixed Integer (Non) Linear Programming strategies that lead to
efficient ways to solve it.
