Circular arcs are the only analytic Jordan curves with an exterior power point

TL;DR

This paper proves that for an analytic Jordan curve, a single exterior power point suffices to determine the boundary as a disk, using Riordan matrices to establish this result.

Contribution

It introduces a novel application of Riordan matrices to show that only one exterior power point is needed for analytic Jordan curves.

Findings

A single exterior power point suffices for analytic Jordan curves.

Riordan matrices are effective in analyzing geometric properties of curves.

The result extends understanding of power points and convex bodies.

Abstract

In 1946, J. Rosenbaum proposed a family of problems asking how many power points are needed to ensure that the boundary $c$ of a given convex body is a disk. In this paper, we use Riordan matrices to show that, if this curve $c$ is analytic, then one single exterior power point is sufficient.