Brunn-Minkowski inequality for p-capacity of compact sets
Abstract
We prove the capacitary Brunn-Minkowski inequality for arbitrary nonempty compact sets in ℝn, n≥2, and every 1<p<∞, with no assumption of convexity, connectedness, interior, or boundary regularity. For 1<p<n the inequality is for the variational p-capacity. For p>n it is for the capacity determined by the constant term at infinity of the normalized p-harmonic Green's function with pole at infinity, and for p=n for an outer regularization of this quantity, which agrees with it for n-regular sets and in the plane is a constant multiple of logarithmic capacity. In particular, it holds for Newtonian capacity of all compact sets in ℝn, n≥3, and for logarithmic capacity of all planar compact sets. Previously these inequalities were known for convex sets and, for logarithmic capacity, for planar continua. When both capacities are positive, equality holds for 0<λ<1 if and only if the filled sets, obtained by adjoining the bounded components of the complement, are convex and equal up to translation and dilation. This characterization is new for nonconvex sets, even for Newtonian and logarithmic capacity.
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BibTeX
@misc{akman2026brunnminkowski,
title = {{Brunn-Minkowski inequality for p-capacity of compact sets}},
author = {Murat Akman},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28421},
eprint = {2610.28421},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28421},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Brunn-Minkowski inequality for p-capacity of compact sets
AU - Murat Akman
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28421
UR - https://arxiv.org/abs/2610.28421
AB - We prove the capacitary Brunn-Minkowski inequality for arbitrary nonempty compact sets in ℝ^n, n≥2, and every 1<p<∞, with no assumption of convexity, connectedness, interior, or boundary regularity. For 1<p<n the inequality is for the variational p-capacity. For p>n it is for the capacity determined by the constant term at infinity of the normalized p-harmonic Green's function with pole at infinity, and for p=n for an outer regularization of this quantity, which agrees with it for n-regular sets and in the plane is a constant multiple of logarithmic capacity. In particular, it holds for Newtonian capacity of all compact sets in ℝ^n, n≥3, and for logarithmic capacity of all planar compact sets. Previously these inequalities were known for convex sets and, for logarithmic capacity, for planar continua. When both capacities are positive, equality holds for 0<λ<1 if and only if the filled sets, obtained by adjoining the bounded components of the complement, are convex and equal up to translation and dilation. This characterization is new for nonconvex sets, even for Newtonian and logarithmic capacity.
ER -