The improved isoperimetric inequality and the Wigner caustic of planar ovals

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.