The φ-Brunn-Minkowski inequality.

For strictly increasing concave functions φ<inline-graphic></inline-graphic> whose inverse functions are log-concave, the φ<inline-graphic></inline-graphic>-Brunn-Minkowski inequality for planar convex bodies is established. It is shown that for convex bodies in Rn<inline-graphic></inline-graphic> the φ<inline-graphic></inline-graphic>-Brunn-Minkowski is equivalent to the φ<inline-graphic></inline-graphic>-Minkowski mixed volume inequalities. [ABSTRACT FROM AUTHOR]