This is a follow up to this recent question: Berry Esseen type result for probability density functions
There exists a multidimensional version of the usual Berry-Esseen theorem (for cumulative distribution functions), see for example this paper (which I cannot find online) of 1945 by Bergström "On the central limit theorem in the space Rk, k> 1".
There also exists a local limit version of the Berry-Esseen theorem for probability density functions: see for instance chapter VII of the book "Sums of independent random variables" by Petrov.
I was not able, however, to find a multidimensional version of this local limit theorem. Does such a result appear somewhere in the literature?
Edit (October 22): I've found a partial answer in this paper of Ričardas Zitikis: "A Berry-Esseen bound for multivariate l-estimates with explicit dependence on dimension" http://www.springerlink.com/content/916020285j808858/
In this paper, a bound is provided (Theorem 1.2) but still depends on a universal constant $c$. It would still be great to known an upper bound on $c$.