Abstract

The Brunn-Minkowski theorem says that vol((1−λ)K+λL)1/n\vol\bigl((1-\lambda)K+\lambda L\bigr)^{1/n}, for K,LK,L convex bodies, is a concave function in λ\lambda, and assuming a common hyperplane projection of KK and LL, it was proved that the volume itself is concave. In this paper we study refinements of Brunn-Minkowski inequality, in the sense of `enhancing' the exponent, either when a common projection onto an (n−kn-k)-plane is assumed or for particular families of sets. In the first case, we show that the expected result of concavity for the kk-th root of the volume is not true, although other Brunn-Minkowski type inequalities can be obtained under the (n−kn-k)-projection hypothesis. In the second case, we show that for pp-tangential bodies, the exponent in Brunn-Minkowski inequality can be replaced by 1/p1/p.

Author information

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María A. Hernández Cifre

Dep. de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Murcia, Spain

mhcifre@um.es

Jesús Yepes Nicolás

Dep. de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Murcia, Spain

jesus.yepes@um.es

Keywords

  • Brunn-Minkowski inequality
  • projections
  • p-tangential bodies

Mathematics Subject Classification

52A20, 52A40; 52A39

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Suggested citation

M. A. Hernández Cifre, J. Yepes Nicolás. “Refinements of the Brunn-Minkowski Inequality.” Journal of Convex Analysis 21 (2014), No. 3, 727–743. https://doi.org/10.68381/jca21039

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Published by Heldermann Verlag, 2014. Rights now held by Banach Press LLC.