  We present a private learner for halfspaces over an arbitrary finite domain
$X\subset \mathbb{R}^d$ with sample complexity $mathrm{poly}(d,2^{\log^*|X|})$.
The building block for this learner is a differentially private algorithm for
locating an approximate center point of $m>\mathrm{poly}(d,2^{\log^*|X|})$
points -- a high dimensional generalization of the median function. Our
construction establishes a relationship between these two problems that is
reminiscent of the relation between the median and learning one-dimensional
thresholds [Bun et al.\ FOCS '15]. This relationship suggests that the problem
of privately locating a center point may have further applications in the
design of differentially private algorithms.
  We also provide a lower bound on the sample complexity for privately finding
a point in the convex hull. For approximate differential privacy, we show a
lower bound of $m=\Omega(d+\log^*|X|)$, whereas for pure differential privacy
$m=\Omega(d\log|X|)$.
