PreprintCC BY-NC-SA 4.0arXiv
An isoperimetric problem for Fourier zeros of centrally symmetric convex bodies
Abstract
Given a centrally symmetric convex body, consider the zero set of the Fourier transform of its characteristic function. We study how large the distance from this set to the origin can be when the volume of the body is fixed. A 2009 conjecture of Benguria, Levitin, and Parnovski asserts that the maximum is attained by a Euclidean ball. We disprove it in all dimensions greater than one. In the plane, every regular centrally symmetric polygon with at least twelve sides outperforms the disk, albeit by a tiny margin, and the regular dodecagon is best among them. In dimensions three and higher, no maximiser exists.
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BibTeX
@misc{gomezserrano2026isoperimetric,
title = {{An isoperimetric problem for Fourier zeros of centrally symmetric convex bodies}},
author = {Javier Gómez-Serrano and Michael Levitin and Daniel Platt and Iosif Polterovich},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10517},
eprint = {2609.10517},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10517},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - An isoperimetric problem for Fourier zeros of centrally symmetric convex bodies
AU - Javier Gómez-Serrano
AU - Michael Levitin
AU - Daniel Platt
AU - Iosif Polterovich
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10517
UR - https://arxiv.org/abs/2609.10517
AB - Given a centrally symmetric convex body, consider the zero set of the Fourier transform of its characteristic function. We study how large the distance from this set to the origin can be when the volume of the body is fixed. A 2009 conjecture of Benguria, Levitin, and Parnovski asserts that the maximum is attained by a Euclidean ball. We disprove it in all dimensions greater than one. In the plane, every regular centrally symmetric polygon with at least twelve sides outperforms the disk, albeit by a tiny margin, and the regular dodecagon is best among them. In dimensions three and higher, no maximiser exists.
ER -