Areas associated with a Strictly Locally Convex Curve

- Journal title : Kyungpook mathematical journal
- Volume 56, Issue 2, 2016, pp.583-595
- Publisher : Department of Mathematics, Kyungpook National University
- DOI : 10.5666/KMJ.2016.56.2.583

Title & Authors

Areas associated with a Strictly Locally Convex Curve

Kim, Dong-Soo; Kim, Dong Seo; Kim, Young Ho; Bae, Hyun Seon;

Kim, Dong-Soo; Kim, Dong Seo; Kim, Young Ho; Bae, Hyun Seon;

Abstract

Archimedes showed 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 S of the region bounded by the parabola X and chord AB is four thirds of the area T of triangle . It is well known that the area U formed by three tangents to a parabola is half of the area T of the triangle formed by joining their points of contact. Recently, the first and third authors of the present paper and others proved that among strictly locally convex curves in the plane , these two properties are characteristic ones of parabolas. In this article, in order to generalize the above mentioned property for parabolas we study strictly locally convex curves in the plane satisfying , where and are some functions on the curves. As a result, we present two conditions which are necessary and sufficient for a strictly locally convex curve in the plane to be an open arc of a parabola.

Keywords

triangle;area;parabola;strictly locally convex curve;plane curvature;

Language

English

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