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

For convex bodies \(K_1,\dots,K_{n-1}\subset{\mathbb R}^n\), their mixed body is defined as the unique (up to translations) convex body \([K_1,\dots,K_{n-1}]\) such that the surface area measures \(S_{n-1}l([K_1,\dots,K_{n-1}];\!\cdot)\!=S(K_1,\dots,K_{n-1};\cdot)\). On the other hand, their mixed projection body \(\Pi(K_1,\dots,K_{n-1})\) is the convex body whose support function is \(h\bigl(\Pi(K_1,\dots,K_{n-1}),u\bigr)=v(K^u_1,\dots,K^u_{n-1})\) for any unit vector \(u\); here \(K^u_i\) is the orthogonal projection of \(K_i\) onto the hyperplane \(u^{\perp}\), and \(v\) denotes the (\(n-1\))-dimensional mixed volume in \(u^{\perp}\).
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:

52A40 Inequalities and extremum problems involving convexity in convex geometry
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