A Note On The Isoperimetric Inequality And Its Stability
Differential Geometry
2011-03-01 v1
Abstract
In this paper, we deals with isoperimetric-type inequalities for closed convex curves in the Euclidean plane R^2. We derive a family of parametric inequalities involving the following geometric functionals associated to a given convex curve with a simple Fourier series proof: length, area of the region included by the curve, area of the domain enclosed by the locus of curvature centers and integral of the radius of curvature. By using our isoperimetric-type inequalities, we also obtain some new geometric Bonnesen-type inequalities. Furthermore we investigate stability properties of such inequalities (near equality implies curve nearly circular).
Cite
@article{arxiv.1102.5642,
title = {A Note On The Isoperimetric Inequality And Its Stability},
author = {Xiang Gao},
journal= {arXiv preprint arXiv:1102.5642},
year = {2011}
}