Area properties associated with strictly convex curves

Bull. Korean Math. Soc. Published online November 4, 2021

Shin-Ok Bang, Dong-Soo Kim, and Incheon Kim
Chonnam National University; Chonnam National University; Chonnam National University

Abstract : Archimedes proved that for a point $P$ on a parabola $X$ and a chord $AB$ of $X$ parallel to the tangent of $X$ at $P$,
the area of the region bounded by the parabola $X$ and chord $AB$ is four thirds of
the area of triangle $\bigtriangleup ABP$. This property was proved to be a characteristic of parabolas, so called the Archimedean
characterization of parabolas.

In this article, we study strictly convex curves in the plane
${\mathbb R}^{2}$. As a result, first using a functional equation we establish a characterization theorem for quadrics.
With the help of this characterization we give another proof of the Archimedean
characterization of parabolas. Finally,
we present two related conditions which are necessary and sufficient for
a strictly convex curve in the plane to be an open arc of a parabola.