The author first introduces a far-reaching generalization of Minkowski’s projection body. For a unit vector u in Euclidean vector space \({\mathbb{R}}^ d\) let \(P_ u\) denote the orthogonal projection on to the \((d-1)-\)dimensional linear subspace \(L_ u\) orthogonal to u. If M is an (\({\mathcal H}^ m,m)\)-rectifiable and \({\mathcal H}^ m\)-measurable subset of \({\mathbb{R}}^ d\), where \(0<m<d\) and \({\mathcal H}^ m\) denotes the m- dimensional Hausdorff measure, then \[ h(\Pi^ m_ M,u)={1\over2}\int_{L_ u}{\mathcal H}^ 0(P_ u^{-1}(y)\cap M)d{\mathcal H}^ m(y) \] defines the support function of a convex body \(\Pi^ m_ M\). This body can be written as an integral of \((d-m)-\)dimensional balls and hence is a zonoid. If K is a d-dimensional convex body and \(\partial K\) its boundary, then \(\Pi_{\partial K}^{d-1}\) is the classical projection body of K. By means of projections on lines and by applying Minkowski’s theorem on the existence of convex bodies with given area function, further auxiliary convex bodies are associated with M. Next, the author proves a very general Poincaré type formula of translative integral geometry and shows that in special cases the results can be expressed by means of mixed volumes involving the projection bodies introduced before. For example, if \(M_ 1,...,M_ m (2\leq m\leq d)\) are \(({\mathcal H}^{d-1},d-1)-\)rectifiable and \({\mathcal H}^{d-1}\)-measurable and satisfy some mild technical condition (which in the interesting cases is satisfied) then \[ \int...\int {\mathcal H}^{d-m}(M_ 1\cap(M_ 2+t_ 2)\cap...\cap(M_ m+t_ m))d\lambda^ d(t_ 2)...d\lambda^ d(t_ m)= \]
\[ =c_{d,m}V(\Pi_{M_ 1}^{d-1},...,\Pi_{M_ m}^{d- 1},B,...,B). \] Here, the integrations are over \({\mathbb{R}}^ d\) with respect to Lebesgue measure \(\lambda^ d\), B is the unit ball in \({\mathbb{R}}^ d\), and \(c_{d,m}\) is a numerical constant. These and similar formulae include and considerably generalize several special cases scattered over the literature. The paper concludes with extremal problems, where known inequalities for mixed volumes are used, and with characterizations of sets of constant brightness. The author announces applications of his results to stochastic geometry, in particular to random hypersurfaces.