  We revisit the problem of finding a minimum enclosing ball with differential
privacy: Given a set of $n$ points in the Euclidean space $\mathbb{R}^d$ and an
integer $t\leq n$, the goal is to find a ball of the smallest radius $r_{opt}$
enclosing at least $t$ input points. The problem is motivated by its various
applications to differential privacy, including the sample and aggregate
technique, private data exploration, and clustering.
  Without privacy concerns, minimum enclosing ball has a polynomial time
approximation scheme (PTAS), which computes a ball of radius almost $r_{opt}$
(the problem is NP-hard to solve exactly). In contrast, under differential
privacy, until this work, only a $O(\sqrt{\log n})$-approximation algorithm was
known.
  We provide new constructions of differentially private algorithms for minimum
enclosing ball achieving constant factor approximation to $r_{opt}$ both in the
centralized model (where a trusted curator collects the sensitive information
and analyzes it with differential privacy) and in the local model (where each
respondent randomizes her answers to the data curator to protect her privacy).
  We demonstrate how to use our algorithms as a building block for
approximating $k$-means in both models.
