The memory centre

Let $x \in \R$ be given. As we know the, amount of bits needed to binary code $x$ with given accuracy ($h \in \R$) is approximately $\m_{h}(x) \approx \log_{2}(\max {1, |\frac{x}{h}|}).$ We consider the problem where we should translate the origin $a$ so that the mean amount of bits needed to code randomly chosen element from a realization of a random variable $X$ is minimal. In other words, we want to find $a \in \R$ such that $$\R \ni a \to \mathrm{E} (\m_{h} (X-a))$$ attains minimum.