Given a positive integer $k$, is there a positive real number $c(k)$ such that $\mathbb{E}\left(\Pi_{i=1}^k X_i\right)\geq c(k)$ 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 $c(k)$? It's known that $c(2)=\frac{1}{6}$.

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

Given a positive integer $k$, is there a positive real number $c(k)$ such that $E(\Pi_{i=1}^k X_i)\geq c(k)$ 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 $c(k)$? It's known that $c(2)=\frac{1}{6}$.

Given a positive integer $k$, is there a positive real number $c(k)$ such that $\mathbb{E}\left(\Pi_{i=1}^k X_i\right)\geq c(k)$ 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 $c(k)$? It's known that $c(2)=\frac{1}{6}$.

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 $c(2)=\frac{1}{6}$.

Given a positive integer $k$, is there a positive real number $c(k)$ such that $E(\Pi_{i=1}^k X_i)\geq c(k)$ 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 $c(k)$? It's known that $c(2)=\frac{1}{6}$.