Five-Point Hyperbolas on Power Curves and Near Lamé-Oval Vertices

Abstract

Problem A33 of Croft--Falconer--Guy records Reznick's question about infinite plane sets for which every five-point subset determines an ellipse or, respectively, a hyperbola, and suggests that $|x|^{2.001}+|y|^{2.001}=1$ might yield only ellipses. We give two results. First, if $p>2$ and $a>0$, every five distinct points of the power curve $u=ay^p$, $y>0$, determine a nondegenerate hyperbola; bounded subarcs therefore give nonconic rectifiable examples for the hyperbolic side of Reznick's question. Second, every fixed one-sided five-point profile, contracted toward an axial vertex of the Lamé oval $|x|^p+|y|^p=1$, eventually determines a nondegenerate hyperbola. Consequently, the suggested Lamé oval with $p=2.001$ has five-point subsets that determine nondegenerate hyperbolas, so it does not yield only ellipses. Symmetric profiles straddling the same vertex, however, determine ellipses. We also separate these results from the classical osculating-conic criterion supplied by equi-affine curvature.