## Return to Question

3 clarified further and made link better; deleted 2 characters in body; deleted 2 characters in body

$H$ is the binary (Shannon) entropyfunction.

In terms of the positive real number $t$, what distribution(s) $\hspace{.01 in}X$ on $\:[0\hspace{.005 in},\hspace{-0.03 in}\scriptsize+\normalsize\infty\hspace{-0.02 in})$
with mean $1$ will minimize the value of the following (one-player) game?

Player chooses a real number $s$ such that $\: 0 < s \leq t \:$ and a
finite partition $\:\left\langle B_0,...,B_n \right\rangle\:$ of $\:[0\hspace{.005 in},\hspace{-0.03 in}\scriptsize+\normalsize\infty\hspace{-0.02 in})^2$ into universally measurable sets
and a real number $s$ such that $\: 0 < s \leq t \;\;$.
.

Have $\;\;\; f \: : \: [0\hspace{.005 in},\hspace{-0.03 in}\scriptsize+\normalsize\infty\hspace{-0.02 in})^2 \: \to \: \{0\hspace{.005 in},...,\hspace{-0.01 in}n\} \;\;\;$ be given by $\;\; \langle x,\hspace{-0.03 in}y\rangle \: \in \: B_{f(\langle x,y\rangle)} \;\;\;\;$.;\;\;$. Challanger samples$x$and$y$independently from$X$, samples$b$uniformly from $\{0\hspace{.005 in},\hspace{-0.04 in}1\}$,$\:$then gives Player$\langle w,\hspace{-0.01 in}z\hspace{.01 in}\rangle \;\; := \;\; \begin{cases} \langle x,y+s\rangle & \text{if } \: b=0 \\ \langle x+s,y\rangle & \text{if } \: b=1 \end{cases}$Player's score is$\:-\big(H\big(b \: | \; f(\langle w,\hspace{-0.01 in}z\hspace{.01 in}\rangle\big)\big) \;\;$. 2 made definition more accurate$H$is the binary entropy function. In terms of the positive real number$t$, what distribution(s)$\hspace{.01 in}X$on$\:[0\hspace{.005 in},\hspace{-0.03 in}\scriptsize+\normalsize\infty\hspace{-0.02 in})$with mean$1$will minimize the value of the following (one-player) game? Player chooses a universally measurable subset finite partition$B$\:\left\langle B_0,...,B_n \right\rangle\:$ of $\:[0\hspace{.005 in},\hspace{-0.03 in}\scriptsize+\normalsize\infty\hspace{-0.02 in})^2$ into universally measurable sets
and a real number $s$ such that $\: 0 < s \leq t \;\;$.