  This paper analyzes the inverse problem of deautoconvolution in the
multi-dimensional case with respect to solution uniqueness and ill-posedness.
Deautoconvolution means here the reconstruction of a real-valued $L^2$-function
with support in the $n$-dimensional unit cube $[0,1]^n$ from observations of
its autoconvolution either in the full data case (i.e. on $[0,2]^n$) or in the
limited data case (i.e. on $[0,1]^n$). Based on multi-dimensional variants of
the Titchmarsh convolution theorem due to Lions and Mikusi\'{n}ski, we prove in
the full data case a twofoldness assertion, and in the limited data case
uniqueness of non-negative solutions for which the origin belongs to the
support. The latter assumption is also shown to be necessary for any uniqueness
statement in the limited data case. A glimpse of rate results for regularized
solutions completes the paper.
