  Solving geometric optimization problems over uncertain data have become
increasingly important in many applications and have attracted a lot of
attentions in recent years. In this paper, we study two important geometric
optimization problems, the $k$-center problem and the $j$-flat-center problem,
over stochastic/uncertain data points in Euclidean spaces. For the stochastic
$k$-center problem, we would like to find $k$ points in a fixed dimensional
Euclidean space, such that the expected value of the $k$-center objective is
minimized. For the stochastic $j$-flat-center problem, we seek a $j$-flat
(i.e., a $j$-dimensional affine subspace) such that the expected value of the
maximum distance from any point to the $j$-flat is minimized. We consider both
problems under two popular stochastic geometric models, the existential
uncertainty model, where the existence of each point may be uncertain, and the
locational uncertainty model, where the location of each point may be
uncertain. We provide the first PTAS (Polynomial Time Approximation Scheme) for
both problems under the two models. Our results generalize the previous results
for stochastic minimum enclosing ball and stochastic enclosing cylinder.
