  Spaces of convex and concave functions appear naturally in theory and
applications. For example, convex regression and log-concave density estimation
are important topics in nonparametric statistics. In stochastic portfolio
theory, concave functions on the unit simplex measure the concentration of
capital, and their gradient maps define novel investment strategies. The
gradient maps may also be regarded as optimal transport maps on the simplex. In
this paper we construct and study probability measures supported on spaces of
concave functions. These measures may serve as prior distributions in Bayesian
statistics and Cover's universal portfolio, and induce distribution-valued
random variables via optimal transport. The random concave functions are
constructed on the unit simplex by taking a suitably scaled (mollified, or
soft) minimum of random hyperplanes. Depending on the regime of the parameters,
we show that as the number of hyperplanes tends to infinity there are several
possible limiting behaviors. In particular, there is a transition from a
deterministic almost sure limit to a non-trivial limiting distribution that can
be characterized using convex duality and Poisson point processes.
