Before revision: Our goal is to resolve a problem proposed by Karatzas and Fernholz  (2008) : Characterizing  the minimum amount of initial capital that would guarantee the investor to beat the market portfolio with a certain probability  as a function of the market configuration and time to maturity.
After revision: Our goal is to resolve a problem proposed by Karatzas and Fernholz On optimal arbitrage (2008) : Characterizing the minimum amount of initial capital that would guarantee the investor to beat the market portfolio with a certain probability as a function of the market configuration and time to maturity.