Abstract
The classical isoperimetric inequality in the Euclidean plane R2 states that for a simple closed curve M of the length LM, enclosing a region of the area AM, 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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
