## 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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