# Existence of moment-constrained maximum entropy distribution with support $[0,1]^n$

Given a finite set of moment values $\{\mu_1,\ldots,\mu_N\}$, for which the multi-dimensional finite Hausdorff moment problem is determined. That is, we know that at least one distribution $\mathcal{D}$ with support $[0,1]^n$ and moments $\mu_i=\int_0^1\cdots\int_0^1 x_1^{r_1}\cdots x_n^{r_n}\,\text{d} \mathcal{D}, \forall i\in \{1,\ldots,N\}$ exits. Here, $r_1+\ldots r_n\leq k\in \mathbb{N}$, i.e. we consider all moments of order at most $k$, e.g. $k=N$ for the one-dimensional case $n=1$.

Is this sufficient for the existence of the maximum differential entropy distribution with the same moments $\{\mu_0,\ldots,\mu_N\}$ and support $[0,1]^n$?

The answer is yes in the one-dimensional if we assume the existing distribution lies in the inner of the moment space. Does this also hold for the multi-dimensional case?

• what is $i$ and what is $r$ in the formula for the moments? – Fedor Petrov Apr 22 '18 at 20:13