An Area Inequality for Ellipses Inscribed in Quadrilaterals

Details An Area Inequality for Ellipses Inscribed in Quadrilaterals An Area Inequality for Ellipses Inscribed in Quadrilaterals

2009-07-09

arxiv.org/abs/0907.1647v1

Journal of Mathematical Inequalities, Volume 4, Issue 3(2010), 431-443

Mathematics

Classical Analysis and ODEs

Submitted for publication to the Journal of Mathematical Inequalities. 16 pages, no figures

Scientific paper

If E is any ellipse inscribed in a convex quadrilateral, D, then we prove that Area(E)/Area(D) is less than or equal to pi/4, and equality holds if and only if D is a parallelogram and E is tangent to the sides of D at the midpoints. This extends well known results for ellipses inscribed in triangles. We also prove that the foci of the unique ellipse of maximal area inscribed in a parallelogram, D, lie on the orthogonal least squares line for the vertices of D. This does not hold in general for convex quadrilaterals.

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