  In stochastic portfolio theory, a relative arbitrage is an equity portfolio
which is guaranteed to outperform a benchmark portfolio over a finite horizon.
When the market is diverse and sufficiently volatile, and the benchmark is the
market or a buy-and-hold portfolio, functionally generated portfolios
introduced by Fernholz provide a systematic way of constructing relative
arbitrages. In this paper we show that if the market portfolio is replaced by
the equal or entropy weighted portfolio among many others, no relative
arbitrages can be constructed under the same conditions using functionally
generated portfolios. We also introduce and study a shaped-constrained
optimization problem for functionally generated portfolios in the spirit of
maximum likelihood estimation of a log-concave density.
