Let \(H_ u\) denote an arbitrary (n-1)-subspace of \({\mathbb{R}}^ n(n>2)\), orthogonal to the direction u. If we write \(K^ n\) for the set of n- dimensional convex bodies and \(\Pi^ n\) for the set of n-zonoids, thenC. M. Petty [Proc. Colloq. Convexity, Copenhagen 1965, 234-241 (1967; Zbl 0152.206)] andR. Schneider [Math. Z. 101, 71-82 (1967; Zbl 0173.247)] have shown that for \(K\in K^ n\) and \(L\in \Pi^ n\) the implication \[ V_{d-1}(K| H_ u)< V_{d-1}(L| H_ u)\quad for\quad all\quad directios\quad u\quad \Rightarrow \quad V_ d(K)< V_ d(L) \] holds, where \(V_ i\) denotes the i-dimensional Lebesgue measure and \(K| H_ u\) is the image of the orthogonal projection of \(K\in K^ n\) onto \(H_ u\). The main result of the present paper is a generalization of a ‘dual’ statement. Namely, let \(S^ n\) denote the set of n-dimensional bodies which are star-shaped with respect to the origin. The intersection body IK of \(K\in S^ n\) is the star body defined by \(r(IK,u)=V_{d-1}(K\cap H_ u)\) for each direction u, in which \(r(M,u)=\max \{\lambda \geq 0:\) \(\lambda\) \(u\in M\}\) is the usual radius function of a compact set M in \({\mathbb{R}}^ n\). If \(I^ n\) denotes the set of intersection bodies, then for \(K\in I^ n\) and \(L\in S^ n\) the implication \[ IK\subset IL\quad \Rightarrow \quad V_ d(K)\leq V_ d(L) \] is shown, where \(V_ d(K)=V_ d(L)\) only if \(K=L.\)
On the other hand, if K is a star body not symmetric about the origin, then there exists a star body L with \[ V_{d-1}(K\cap H_ u)<V_{d- 1}(L\cap H_ u)\quad for\quad all\quad directions\quad u\quad \Rightarrow \quad V_ d(K)>V_ d(L). \] For proving these and other interesting results, the author uses dual mixed volumes as a suitable method for solving problems involving intersections. (Conversely, for projection problems the use of mixed volumes is a well-known, powerful method.)
Therefore one can find also extensive, informative material to several topics concerning mixed volumes (e.g. projection bodies and the Petty- Schneider result) which are compared with corresponding subjects related to dual mixed volumes (e.g. intersection bodies and a known problem of Busemann and Petty which seemingly gave the main motivation for that paper).
From this viewpoint, the paper might be considered as a useful survey on various important tools from geometric convexity, too.

Reviewer: H.Martini

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