Hölder Duality and Self-Dual Arc Length on Lp Unit Curves
Abstract
We study arc length on the first-quadrant unit curve xp+yp=1, p>1, when length is measured with respect to an Lr norm. Denoting this length by L(p,r), we prove the Hölder-duality identity L(p,r)=L(r*,p*), where p*=p/(p−1) and r*=r/(r−1). The identity is interpreted as a special case of the symmetry of mixed area (equivalently, of the anisotropic perimeter pairing). For the distinguished dual metric r=p*, the length element agrees with the areal angular element: L(p,p*)=∫γ(x dy−y dx)=1/p B(1/p,1/p). Thus the familiar coincidence between arc length and angular parameter on the Euclidean unit circle is the self-dual case p=2 of a more general relation between Lp geometry and its dual metric. As a consequence, the Euclidean arc length of the Lp unit curve admits a dual trigonometric representation.
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BibTeX
@misc{concha2026hölder,
title = {{Hölder Duality and Self-Dual Arc Length on $L^p$ Unit Curves}},
author = {Jacinto Joaquín Arichabala Concha},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03693},
eprint = {2611.03693},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03693},
}
RIS
TY - UNPB
TI - Hölder Duality and Self-Dual Arc Length on L^p Unit Curves
AU - Jacinto Joaquín Arichabala Concha
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.03693
UR - https://arxiv.org/abs/2611.03693
AB - We study arc length on the first-quadrant unit curve x^p+y^p=1, p>1, when length is measured with respect to an L^r norm. Denoting this length by L(p,r), we prove the Hölder-duality identity L(p,r)=L(r^*,p^*), where p^*=p/(p−1) and r^*=r/(r−1). The identity is interpreted as a special case of the symmetry of mixed area (equivalently, of the anisotropic perimeter pairing). For the distinguished dual metric r=p^*, the length element agrees with the areal angular element: L(p,p^*)=∫_γ(x dy−y dx)=1/p B(1/p,1/p). Thus the familiar coincidence between arc length and angular parameter on the Euclidean unit circle is the self-dual case p=2 of a more general relation between L^p geometry and its dual metric. As a consequence, the Euclidean arc length of the L^p unit curve admits a dual trigonometric representation.
ER -