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

Zhao, Chang-Jian

Bull. Belg. Math. Soc. - Simon Stevin 22, No. 3, 391-401 (2015).

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.

Reviewer: Alina Stancu (Montréal)

MSC:

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

Keywords:

dual Brunn-Minkowski inequality; harmonic Blaschke addition; radial Blaschke addition; radial Minkowski addition