PreprintCC BY 4.0arXiv
Improved Brunn--Minkowski inequality for θ-convolution bodies
Abstract
For n≥ 2, we study the optimal function φn in the Brunn-Minkowski inequality for θ-convolution bodies |K+θL|1/n≥φn(θ)(|K|1/n+|L|1/n) for any convex bodies K,L⊂ℝn. We improve the known lower bound for every 0<θ<(3/4)n and determine φn on an interval strictly containing [(2/3)n,1]. For θ<1 in this interval, equality holds if and only if K and −L are translates of the same n-dimensional simplex. The proof reduces the geometric problem to a sharp one-dimensional estimate for nonnegative concave functions under a weighted integral constraint.
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BibTeX
@misc{goni2026improved,
title = {{Improved Brunn--Minkowski inequality for $θ$-convolution bodies}},
author = {Javier Martín Goñi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37066},
eprint = {2610.37066},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37066},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Improved Brunn--Minkowski inequality for θ-convolution bodies
AU - Javier Martín Goñi
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37066
UR - https://arxiv.org/abs/2610.37066
AB - For n≥ 2, we study the optimal function φ_n in the Brunn-Minkowski inequality for θ-convolution bodies |K+_θL|^1/n≥φ_n(θ)(|K|^1/n+|L|^1/n) for any convex bodies K,L⊂ℝ^n. We improve the known lower bound for every 0<θ<(3/4)^n and determine φ_n on an interval strictly containing [(2/3)^n,1]. For θ<1 in this interval, equality holds if and only if K and −L are translates of the same n-dimensional simplex. The proof reduces the geometric problem to a sharp one-dimensional estimate for nonnegative concave functions under a weighted integral constraint.
ER -