  This paper introduces a new way to compact a continuous probability
distribution $F$ into a set of representative points called support points.
These points are obtained by minimizing the energy distance, a statistical
potential measure initially proposed by Sz\'ekely and Rizzo (2004) for testing
goodness-of-fit. The energy distance has two appealing features. First, its
distance-based structure allows us to exploit the duality between powers of the
Euclidean distance and its Fourier transform for theoretical analysis. Using
this duality, we show that support points converge in distribution to $F$, and
enjoy an improved error rate to Monte Carlo for integrating a large class of
functions. Second, the minimization of the energy distance can be formulated as
a difference-of-convex program, which we manipulate using two algorithms to
efficiently generate representative point sets. In simulation studies, support
points provide improved integration performance to both Monte Carlo and a
specific Quasi-Monte Carlo method. Two important applications of support points
are then highlighted: (a) as a way to quantify the propagation of uncertainty
in expensive simulations, and (b) as a method to optimally compact Markov chain
Monte Carlo (MCMC) samples in Bayesian computation.
