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Width-integrals and affine surface area of convex bodies. (English) Zbl 1155.52005

For convex bodies K 1 ,,K n-1 n , their mixed body is defined as the unique (up to translations) convex body [K 1 ,,K n-1 ] such that the surface area measures S n-1 l([K 1 ,,K n-1 ];·)=S(K 1 ,,K n-1 ;·). On the other hand, their mixed projection body Π(K 1 ,,K n-1 ) is the convex body whose support function is hΠ ( K 1 , , K n-1 ) , u=v(K 1 u ,,K n-1 u ) for any unit vector u; here K i u is the orthogonal projection of K i onto the hyperplane u , and v denotes the (n-1)-dimensional mixed volume in u .

In this paper the authors obtain Brunn-Minkowski type inequalities for the width-integrals of of mixed projection bodies and for the affine surface area of mixed bodies.

MSC:
52A40Inequalities and extremum problems (convex geometry)