On the Weyl tube formula for nonlocal perimeters
Abstract
We study the asymptotic behaviour of the nonlocal perimeter of tubular neighbourhoods. For a closed C1,1 submanifold of Euclidean space, we show that a suitable rescaling of the nonlocal perimeter of its tube converges to an explicit constant times the Hausdorff measure of the submanifold. The constant is the fractional perimeter of the unit ball in the codimension of the submanifold. For finite point sets, we obtain an exact tube formula in terms of a generalised hypergeometric function. In contrast with the classical Weyl formula, this expansion has infinitely many terms, contains both noninteger and integer powers of the tubular radius, and has coefficients that depend on the relative positions of the points rather than only on their intrinsic geometry. We also establish a point-set expansion for general analytic kernels.
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BibTeX
@misc{dipierro2026weyl,
title = {{On the Weyl tube formula for nonlocal perimeters}},
author = {Serena Dipierro and Jack Thompson and Enrico Valdinoci},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16461},
eprint = {2610.16461},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16461},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the Weyl tube formula for nonlocal perimeters
AU - Serena Dipierro
AU - Jack Thompson
AU - Enrico Valdinoci
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16461
UR - https://arxiv.org/abs/2610.16461
AB - We study the asymptotic behaviour of the nonlocal perimeter of tubular neighbourhoods. For a closed C^1,1 submanifold of Euclidean space, we show that a suitable rescaling of the nonlocal perimeter of its tube converges to an explicit constant times the Hausdorff measure of the submanifold. The constant is the fractional perimeter of the unit ball in the codimension of the submanifold. For finite point sets, we obtain an exact tube formula in terms of a generalised hypergeometric function. In contrast with the classical Weyl formula, this expansion has infinitely many terms, contains both noninteger and integer powers of the tubular radius, and has coefficients that depend on the relative positions of the points rather than only on their intrinsic geometry. We also establish a point-set expansion for general analytic kernels.
ER -