Monotonicity and concavity of integral functionals involving area measures of convex bodies

2016/02/18 by Colesanti, Andrea, Hug, Daniel, Saorín-Gómez, Eugenia
#52A20 #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1602.05994

Abstract

For a broad class of integral functionals defined on the space of n-dimensional convex bodies, we establish necessary and sufficient conditions for monotonicity, and necessary conditions for the validity of a Brunn-Minkowski type inequality. In particular, we prove that a Brunn-Minkowski type inequality implies monotonicity, and that a general Brunn-Minkowski type inequality is equivalent to the functional being a mixed volume.

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