  We prove that higher moment maps on area measures of a euclidean vector space
are injective, while the kernel of the centroid map equals the image of the
first variation map.
  Based on this, we introduce the space of smooth dual area measures on a
finite-dimensional euclidean vector space and prove that it admits a natural
convolution product which encodes the local additive kinematic formulas for
groups acting transitively on the unit sphere.
  As an application of this new integral-geometric structure, we obtain the
local additive kinematic formulas in hermitian vector spaces in a very explicit
way.
