Optimal Transport on the Probability Simplex with Logarithmic Cost

Gabriel Khan, Jun Zhang

Published 2018-11-30Version 1

Motivated by the financial problem of building financial portfolios which outperform the market, Pal and Wong considered optimal transport on the probability simplex $\triangle^n$ where the cost function is induced by the free energy. We study the regularity of this problem and find that the associated $MTW$ tensor is non-negative definite and in fact constant on $\triangle^n \times \triangle^n$. We further find that relative $c$-convexity corresponds to the standard notion of convexity in the probability simplex. Hence, we are able to use standard results in optimal transport to establish regularity for the optimal transport maps considered by Pal and Wong. We also provide several new examples of costs satisfying the $MTW(0)$ condition.