2015/12/31 by Michał Zwierzyński · 1 citation
Mathematics · #Affine transformation #Caustic (mathematics) #Constant (computer programming) #Equidistant #Euclidean geometry #Geometric Analysis and Curvature Flows #Geometry #Isoperimetric dimension #Isoperimetric inequality #Mathematical analysis #Mathematics #Mathematics and Applications #Plane (geometry) #Point processes and geometric inequalities #Pure mathematics #Simple (philosophy) #math.DG
paper · pdf · doi:10.1016/j.jmaa.2016.05.016
15 pages, 4 figures
arxiv created 2016/05/07 · openalex publication_date 2016/05/11 · openalex created_date 2016/06/24 · arxiv updated 2016/07/06 · openalex updated_date 2026/08/05
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 gets LM2\geqslant 4πAM. In this paper we present the improved isoperimetric inequality, which states that if M is a closed regular simple convex curve, then LM2\geqslant 4πAM+8π|\widetildeA_E(1)/(2)(M)|, where \widetildeA_E(1)/(2)(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=(1)/(2)) and the improved isoperimetric inequality is a consequence of certain bounds of oriented areas of affine equidistants.