A Sharp Planar Fractional Isoperimetric Inequality
Abstract
In this article, we prove the sharp form of the fractional isoperimetric inequality in the plane, originally posed by Maz'ya [Problem 1, Integral Equations Operator Theory, 2018]. More precisely, we show that, for any given s∈(0,1) and any bounded domain Ω⊂ ℝ2 with C1 boundary, Ps(Ω) ≤ (πs−1/2Γ((3−s)/2))/(s(1−s)Γ((4−s)/2))[ℋ1(∂Ω)]2−s, where Ps denotes the fractional s-perimeter, Γ denotes the Gamma function, and ℋ1 denotes the $1$-dimensional Hausdorff measure on ℝ2. The constant is sharp and equality is attained by the disk. The proof proceeds in three steps: reducing the fractional perimeter to a chord functional, reducing connected domains to convex bodies, and proving the sharp convex chord inequality via a fractional Willmore-type inequality, a variational formula along outer parallel bodies, and asymptotic analysis.
Related
- The Sharp Isoperimetric Inequality in the First Heisenberg Group
- The sharp isoperimetric inequality for varifolds in codimension two
- Sharp concentration levels and extremal functions for a planar Choquard-type Trudinger--Moser inequality
- The fractional Faber--Krahn inequality in sharp quantitative form
- The Isoperimetric Inequality in Higher-Dimensional Heisenberg Groups
BibTeX
@misc{lin2026sharp,
title = {{A Sharp Planar Fractional Isoperimetric Inequality}},
author = {Xiaosheng Lin and Dachun Yang and Sibei Yang and Wen Yuan and Yangyang Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.19052},
eprint = {2609.19052},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19052},
}
RIS
TY - UNPB
TI - A Sharp Planar Fractional Isoperimetric Inequality
AU - Xiaosheng Lin
AU - Dachun Yang
AU - Sibei Yang
AU - Wen Yuan
AU - Yangyang Zhang
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2609.19052
UR - https://arxiv.org/abs/2609.19052
AB - In this article, we prove the sharp form of the fractional isoperimetric inequality in the plane, originally posed by Maz'ya [Problem 1, Integral Equations Operator Theory, 2018]. More precisely, we show that, for any given s∈(0,1) and any bounded domain Ω⊂ ℝ^2 with C^1 boundary, P_s(Ω) ≤ (π^s−1/2Γ((3−s)/2))/(s(1−s)Γ((4−s)/2))[ℋ^1(∂Ω)]^2−s, where P_s denotes the fractional s-perimeter, Γ denotes the Gamma function, and ℋ^1 denotes the $1$-dimensional Hausdorff measure on ℝ^2. The constant is sharp and equality is attained by the disk. The proof proceeds in three steps: reducing the fractional perimeter to a chord functional, reducing connected domains to convex bodies, and proving the sharp convex chord inequality via a fractional Willmore-type inequality, a variational formula along outer parallel bodies, and asymptotic analysis.
ER -