A refined reverse isoperimetric inequality in the plane

Abstract

Abstract. It is proved that if γ is a closed strictly convex curve in the plane with length L and area A , then L 2 4πA + 2π|Ã|, with equality holding if and only if γ is a circle, whereà denotes the oriented area enclosed by the locus of curvature centers of γ .Mathematics subject classification (2010): 52A38, 52A40.

A Note on the Reverse Isoperimetric Inequality

“…It follows from the proof that for an arbitrarily ε < 1 4 , Φ(ε) can not always be non-negative for arbitrarily curve γ, which confirms the conjecture by Pan et al [2]: Corollary 3.2. The best constant ε in the inequality L 2 ≤ 4πA + ε|Ã| is π.…”

“…The reverse isoperimetric inequality (1.2) is not sharp actually, thus Pan et al [2] provide an improved version of (1.2) under the same assumptions of Theorem 1.2:…”

“…In Sect. 3, we will provide a simpler proof of Theorem 1.3 by using Fourier series, which is different from the approach in [2]. In Sect.…”

ON BONNESEN-STYLE ALEKSANDROV-FENCHEL INEQUALITIES IN ℝn

“…Only a few upper bounds for convex domains are known (see [13,29,28,35,48]) and an upper bound for an oval domain in R 2 was given by Bottema. The higher-dimensional cases are more complicated.…”

Some remarks about closed convex curves

“…Among them the isoperimetric inequalities are of special interest; see [Ball 1991;Blaschke 1956;Bonnesen 1929;Osserman 1978;1979;Pan and Zhang 2007;Schneider 1993] and references therein. For convex curves in the Euclidean plane ‫ޒ‬ 2 , there are also many interesting inequalities involving their geometric quantities such as inradius, outradius, width, area, length and curvature or radius of curvature; see for example [Chernoff 1969;Gage 1983;Green and Osher 1999;Hernández Cifre 2000;Ma and Cheng 2009;Ma and Zhu 2008;Pan and Yang 2008;Sholander 1952].…”