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Width-integrals and affine surface area of convex bodies. (English) Zbl 1155.52005
For convex bodies $K_1,\dots,K_{n-1}\subset{\Bbb R}^n$, 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.
52A40Inequalities and extremum problems (convex geometry)
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