Brunn-Minkowski inequality for θ-convolution bodies via Ball's bodies

2022/11/30 by Alonso-Gutiérrez, David, Goñi, Javier Martín
#52A40 #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2211.17069

Abstract

We consider the problem of finding the best function φ_n:[0,1]→ℝ such that for any pair of convex bodies K,L∈ℝⁿ the following Brunn-Minkowski type inequality holds |K+_θL|^(1)/(n)≥φ_n(θ)(|K|^(1)/(n)+|L|^(1)/(n)), where K+_θL is the θ-convolution body of K and L. We prove a sharp inclusion of the family of Ball's bodies of an α-concave function in its super-level sets in order to provide the best possible function in the range ((3)/(4))ⁿ≤θ≤1, characterizing the equality cases.

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