Hölder Duality and Self-Dual Arc Length on $L^p$ Unit Curves

Abstract

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: $L(p,p^*)=\int_γ(x\,dy-y\,dx)=\frac1p B(1/p,1/p)$. 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.