The reverse isoperimetric inequality for convex plane curves through a length-preserving flow
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Archiv der Mathematik
The reverse isoperimetric inequality for convex plane curves through a length-preserving flow Yunlong Yang
and Weiping Wu
Abstract. By a length-preserving flow, we provide a new proof of a conjecture on the reverse isoperimetric inequality composed by Pan et al. (Math Inequal Appl 13:329–338, 2010), which states that if γ is a convex curve with length L and enclosed area A, then the best constant ε in the inequality ˜ L2 ≤ 4πA + ε|A| is π, where A˜ denotes the oriented area of the locus of its curvature centers. Mathematics Subject Classification. 52A40, 52A10, 53C44. Keywords. Convex plane curves, Length-preserving flow, Reverse isoperimetric inequality.
1. Introduction. The classical isoperimetric inequality states that for a simple closed plane curve γ of length L and enclosed area A, one has L2 − 4πA ≥ 0, and the equality holds if and only if γ is a circle. There have been many proofs, sharpened forms, generalizations, and applications of this inequality, see, for instance, [1,3,4,7,12] and the literature therein. Let γ be a closed strictly convex plane curve with length L and enclosed area A. In [10], Pan and Zhang established a reverse isoperimetric inequality, that is, ˜ L2 ≤ 4π(A + |A|), This work was supported by the Fundamental Research Funds for the Central Universities (Nos. 3132020172, 3132019177).
Y. Yang and W. Wu
Arch. Math.
where A˜ is the oriented area enclosed by the locus of centers of curvature of γ, and the equality holds if and only if γ is a circle. Pan et al. [8] improved the above result to ˜ L2 ≤ 4πA + 2π|A|, and they also conjectured that the best constant ε in the inequality ˜ L2 ≤ 4πA + ε|A|
(1.1)
would be π. In [5], Gao proved the conjectured inequality and showed that this inequality is sharp. Recently, researching inequalities of plane curves through different geometric flows has attracted increasing interest. By the curve shortening flow, Pankrashkin [11] gave an inequality of a simple plane curve which contains its maximum curvature, and S¨ ussmann [13,14] proved the Banchoff–Pohl inequality for special classes of curves in the plane or that in symmetric Minkowski geometries. Ivaki [6] and Stancu [15] focused on affine inequalities via centroaffine curvature flows. In this short paper, motivated by the work of Pankrashkin [11], we will give another proof of the conjectured reverse isoperimetric inequality (1.1) through the length-preserving flow 1 N, (1.2) γt = h − κ γ(θ, 0) = γ0 (θ),
θ ∈ S1,
(1.3)
where γ is the position vector of the evolving curve, κ and h are its curvature and support function, and N is the unit inner normal of γ. The paper is organized as follows: In Section 2, we give some basic facts about convex plane curves and geometric flows. In Section 3, we give the proof of the conjectured reverse isoperimetric inequality (1.1). 2. Preliminaries. For a convex curve γ, its support function h is defined by h(θ) = −γ(θ), N (θ),
(2.1)
where θ is the normal angle and N (θ) is the unit inward pointing normal vector along this curve. Clearly, h(
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