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Dual cyclic Brunn-Minkowski inequalities. (English) Zbl 1331.52016

An application of Minkowski and Hölder inequalities to functions on the sphere \(\mathbb S^{n-1}\) led the author to a new interpolation type inequality between norms \(L_r\), \(L_s\), \(L_t\) for \(r\), \(s\), \(t \in \mathbb R\). This was furthermore used to derive new Brunn-Minkowski type inequalities for dual quermassintegrals of star bodies in \(\mathbb{R}^n\) considered successively with the Minkowski addition, and radial Blaschke addition, and harmonic Blaschke addition. The author’s new Brunn-Minkowski inequalities generalize known dual Brunn-Minkowski inequalities for the radial additions listed above, two of them dating back to Lutwak’s paper on dual mixed volumes from 1975 where many Brunn-Minkowski duality questions originate.

MSC:

52A40 Inequalities and extremum problems involving convexity in convex geometry
52A30 Variants of convex sets (star-shaped, (\(m, n\))-convex, etc.)
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Full Text: Euclid