Given a positive integer $k$, is there a positive real number $c(k)$ such that  $\int_{[0,1]} \Pi_{i=1}^k X_i dx\geq c(k)$ for any $k$-random variables $X_1,X_2,...,X_k$ which all have the uniform distribution over $[0,1]$?  What's the optimal value of $c(k)$? We know that $c(2)=\frac{1}{6}$.