  In this paper we provide new methodology for inference of the geometric
features of a multivariate density in deconvolution. Our approach is based on
multiscale tests to detect significant directional derivatives of the unknown
density at arbitrary points in arbitrary directions. The multiscale method is
used to identify regions of monotonicity and to construct a general procedure
for the detection of modes of the multivariate density. Moreover, as an
important application a significance test for the presence of a local maximum
at a pre-specified point is proposed. The performance of the new methods is
investigated from a theoretical point of view and the finite sample properties
are illustrated by means of a small simulation study.
