  Differential privacy (DP), provides a framework for provable privacy
protection against arbitrary adversaries, while allowing the release of summary
statistics and synthetic data. We address the problem of releasing a noisy
real-valued statistic vector $T$, a function of sensitive data under DP, via
the class of $K$-norm mechanisms with the goal of minimizing the noise added to
achieve privacy. First, we introduce the sensitivity space of $T$, which
extends the concepts of sensitivity polytope and sensitivity hull to the
setting of arbitrary statistics $T$. We then propose a framework consisting of
three methods for comparing the $K$-norm mechanisms: 1) a multivariate
extension of stochastic dominance, 2) the entropy of the mechanism, and 3) the
conditional variance given a direction, to identify the optimal $K$-norm
mechanism. In all of these criteria, the optimal $K$-norm mechanism is
generated by the convex hull of the sensitivity space. Using our methodology,
we extend the objective perturbation and functional mechanisms and apply these
tools to logistic and linear regression, allowing for private releases of
statistical results. Via simulations and an application to a housing price
dataset, we demonstrate that our proposed methodology offers a substantial
improvement in utility for the same level of risk.
