Abstract
We investigate the isoperimetric deficit of the oval domain in the Euclidean plane. Via the kinematic formulae of Poincaré and Blaschke, and Blaschke’s rolling theorem, we obtain a sharp reverse Bonnesen-style inequality for a plane oval domain, which improves Bottema’s result. Furthermore, we extend the isoperimetric deficit to the symmetric mixed isoperimetric deficit for two plane oval domains, and we obtain two reverse Bonnesen-style symmetric mixed inequalities, which are generalizations of Bottema’s result and its strengthened form.
Highlights
1 Introduction and main results Integral geometry originated from geometric probability
Geometric inequality is an important topic in integral geometry
Perhaps the classical isoperimetric inequality is the oldest geometric inequality, that is, the disc encloses the maximum area among all domains of fixed perimeter
Summary
Introduction and main resultsIntegral geometry originated from geometric probability. For the oval domain K in R2, Bottema obtained the following reverse Bonnesen-style inequality (see [5]): P2 – 4π A ≤ π 2(ρM – ρm)2, (1.5) Zhou et al obtained some reverse Bonnesen-style inequalities for any convex domain in [33].
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