کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
4614028 | 1339278 | 2016 | 14 صفحه PDF | دانلود رایگان |
The classical isoperimetric inequality in the Euclidean plane R2R2 states that for a simple closed curve M of the length LMLM, enclosing a region of the area AMAM, one getsLM2⩾4πAM. In this paper we present the improved isoperimetric inequality, which states that if M is a closed regular simple convex curve, thenLM2⩾4πAM+8π|A˜E12(M)|, where A˜E12(M) is an oriented area of the Wigner caustic of M, and the equality holds if and only if M is a curve of constant width. Furthermore we also present a stability property of the improved isoperimetric inequality (near equality implies curve nearly of constant width). The Wigner caustic is an example of an affine λ -equidistant (for λ=12) and the improved isoperimetric inequality is a consequence of certain bounds of oriented areas of affine equidistants.
Journal: Journal of Mathematical Analysis and Applications - Volume 442, Issue 2, 15 October 2016, Pages 726–739