The log-Brunn-Minkowski inequality in ℝ

Böröczky, Lutwak, Yang, and Zhang have recently proved the log-Brunn-Minkowski inequality for two origin-symmetric convex bodies in the plane which is stronger than the classical Brunn-Minkowski inequality. This paper establishes the log-Brunn-Minkowski, log-Minkowski, L p L_p -Minkowski, and L p L_p -Brunn-Minkowski inequalities for two classes of convex bodies in R 3 \mathbb {R}^3 .