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7

Inequalities for mixed volumes

7.1 The Brunn–Minkowski theorem

Studying the behaviour of volume under Minkowski addition, one is led to the notion of mixed volumes on one hand and to the Brunn–Minkowski theorem on the other. In its simplest form, this theorem says that for two convex bodies , each of volume 1, the Minkowski sum (K + L) has volume at least 1, and its volume is equal to 1 only if K and L are translates. We shall give three proofs, each of which has its merits and illuminates the scenery from a different point of view. The Brunn–Minkowski theorem is the starting point for a rich theory of geometric (and also analytic) inequalities, ...

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