Ptolemy constant and uniformity

Abstract

We study Ptolemy constant and uniformity constant in various plane domains including triangles, quadrilaterals and ellipses.

Ptolemy Constants as Described by Eccentricity

“…for a rectangle of eccentricity ε, which again is the lower bound given in [1], Corollary 4.8. The threshold…”

“…for an ellipse of eccentricity ε, which is the lower bound given in [1], Theorem 1.7. Consider instead vertices v 1 , v 2 , v 3 , v 4 of a convex quadrilateral inscribed within the planar rectangle…”

“…It is thought that P (ε) measures the "roundness of planar curves". Harmaala & Klén [1] proved that 1 2…”