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On the Lp Brunn--Minkowski inequality for planar dual quermassintegrals
Abstract
We determine the sharp Lp Brunn--Minkowski range for planar dual quermassintegrals on origin-symmetric convex bodies: 0<q≤ 2 for 0≤ p<1, and 0<q≤4 for p=1. The main new results are the Minkowski inequality for 2<q<4, proved using a weighted spectral inequality on the projective circle, and parallelogram counterexamples for 0<p<1 and q>2. Together with the known logarithmic and endpoint results, we obtain the complete equality classification: only positive dilates occur, except at (p,q)=(0,2), where parallelograms with corresponding sides parallel also give equality.
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BibTeX
@misc{tao2026brunnminkowski,
title = {{On the $L_p$ Brunn--Minkowski inequality for planar dual quermassintegrals}},
author = {Jiangyan Tao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09636},
eprint = {2610.09636},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09636},
}
RIS
TY - UNPB
TI - On the L_p Brunn--Minkowski inequality for planar dual quermassintegrals
AU - Jiangyan Tao
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09636
UR - https://arxiv.org/abs/2610.09636
AB - We determine the sharp L_p Brunn--Minkowski range for planar dual quermassintegrals on origin-symmetric convex bodies: 0<q≤ 2 for 0≤ p<1, and 0<q≤4 for p=1. The main new results are the Minkowski inequality for 2<q<4, proved using a weighted spectral inequality on the projective circle, and parallelogram counterexamples for 0<p<1 and q>2. Together with the known logarithmic and endpoint results, we obtain the complete equality classification: only positive dilates occur, except at (p,q)=(0,2), where parallelograms with corresponding sides parallel also give equality.
ER -