The memory center

As we know, the length of binary code of a point x∈Rx∈R (with accuracy h > 0) is approximately mh(x)≈log2max1,xh. We will consider the problem where we should translate the origin a of the coordinate system so that the mean amount of bits needed to code a randomly chosen element from a realization of a random variable X   is minimal. In other words, we want to find a∈Ra∈R such thatR∋a→E(mh(X-a))R∋a→E(mh(X-a))attains minimum.We show that under reasonable assumptions the choice of a does not depend on h asymptotically. Consequently, we reduce the problem to finding the minimum of functionR∋a→∫Rln(|x-a|)f(x)dx,R∋a→∫Rln(|x-a|)f(x)dx,where f is the density distribution of the random variable X. Moreover, we provide constructive approach for determining a.