A Short Note on Gage’s Isoperimetric Inequality ∗ Hong Lu Shengliang Pan
Department of Mathematics, East China Normal University, Shanghai, 200062, P. R. China email:
[email protected] December 7, 2004 Abstract In this short note we shall first use Minkowski’s support function to restate Gage’s isoperimetric inequality as an integral inequality about positive periodic function, which can be considered as a mathematical analysis version of Gage’s inequality, and then strengthen the inequality in a more isoperimetric form as a conjecture. We are expecting a proof of this conjecture. M athematics Subject Classif ication : 26D15, 52A38, 52A40 Key words convex plane curves, classical isoperimetric inequality, Gage’s inequality, Minkowski’s support function, integral inequality.
1
Introduction
The classical isoperimetric inequality is probably one of the best known results among all global properties about closed plane curves, which states that for a simple ∗
This work is supported in part by the National Science Foundation of China (10371039), the Shanghai Science and Technology Committee Program and the Shanghai Priority Academic Discipline
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closed curve Γ (in the Euclidean plane) of length L enclosing a region of area A, one gets that L2 − 4πA ≥ 0, (1.1) and the equality holds if and only if Γ is a circle. This inequality was known to the ancient Greeks, mathematical proofs were only given, however, in the 19th century by J. Steiner [18]. There are many proofs, sharpened forms, generalizations, and applications of this inequality, see, e.g., [1], [2], [12], [13], [15], [16], [19], etc., and the literature therein. Usually, various variants of the classical isoperimetric inequality do not involve the integral of curvature of the plane curve. In the 1980’s, however, Gage [3] has shown another “isoperimetric inequality” which involves the integration of the squared curvature, that is, if k is the signed curvature of a closed convex plane curve α with length L and enclosing area A, then Z
k 2 ds ≥
α
πL . A
(1.2)
Gage [3] also presents an example of H. Jacobowitz which shows that inequality (1.2) does not always hold for the bone shaped nonconvex curves. Inequality (1.2) plays an essential role in the studying of the curve-shortening flow in the plane (see [4], [5], [6], [7], and [17]), and is now called by us Gage’s inequality. In [8], there are another proof and some generalizations of Gage’s inequality. In this short note we will first give a mathematical analysis version of Gage’s inequality in terms of the Minkowski support function of a convex closed plane curve, which is an integral inequality about positive periodic function, and then, we shall strengthen Gage’s inequality in a more isoperimetric form as a conjecture. Acknowledgement We would like to thank Professors Huitao Feng, Shanwen Hu, Xuecheng Peng, Chunli Shen, Wanghui Yu and Feng Zhou and other colleagues of our geometric inequalities seminar for their comments, help and encouragement.
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The Polar Tangential Coordinates
Let α be a C 1 regular and positively oriented closed convex curve in the plane, O be a point inside α which is chosen to be the origin of our fixed frame. If p is the oriented perpendicular distance from O to the tangent line Γ at a point (α1 , α2 ) on α, and θ is the oriented angle from the positive half of the x1 –axis to this perpendicular ray to Γ, p is a single-valued function of θ and, therefore, is periodic with period 2π, and the equation of Γ can be written in the form x1 cos θ + x2 sin θ = p(θ). 2
(2.1)
All the tangent lines to α form a one–parameter family of lines, (2.1) is the equation of this family, θ is the parameter. The curve α is then the envelope of this family and, therefore, α can be parametrized as (
α1 = p(θ) cos θ − p0 (θ) sin θ α2 = p(θ) sin θ + p0 (θ) cos θ,
(2.2)
where the prime denotes the derivative with respect to θ. Thus, if θ and p(θ) are given, one can uniquely determine a point (α1 , α2 ) on the curve α, and vice versa. (θ, p(θ)) is usually called the polar tangential coordinate of a point (α1 , α2 ) on α, and p(θ) the Minkowski support function of α. Differentiation of (2.2) with respect to θ gives (
α10 = −[p(θ) + p00 (θ)] sin θ α20 = [p(θ) + p00 (θ)] cos θ
(2.3)
From the definition of θ, the unit tangential vector field of α is given by T~ = (− sin θ, cos θ), therefore one has p(θ) + p00 (θ) > 0.
(2.4)
Since the inner pointing unit normal vector is given by ~ = (− cos θ, − sin θ), N from Frenet formulas, one obtains that the signed curvature k of α is given by dθ . (2.5) ds If L and A are respectively the length of α and the area bounded by α, then, observing that p(θ) is a periodic function with period 2π, one gets k=
L=
Z
2π
q
0
α10 2
+
α20 2 dθ
=
Z
2π
00
[p(θ) + p (θ)]dθ =
Z
0
2π
p(θ)dθ
(2.6)
0
which is known as Cauchy’s formula, and A = 1/2
Z α
α1 dα2 − α2 dα1 = 1/2
Z
2π
p(θ)[p(θ) + p00 (θ)]dθ
(2.7)
0
which is known as Blaschke’s formula. Now, in order to express the signed curvature k of α in terms of the support function, one needs the additional assumption that α is of class C 2 and differentiates (2.3) with respect to θ once more to get k=
α10 α200 − α100 α20 1 = > 0. 02 0 2 3/2 p(θ) + p00 (θ) [α1 + α2 ] 3
(2.8)
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Restatement of Gage’s Isoperimetric Inequality and Some Remarks
Since the total curvature of a simple closed and positively oriented plane curve α is R 2π, that is, α kds = 2π, from Cauchy-Schwartz inequality, one gets that Z
k 2 ds ≥
α
4π 2 . L
(3.1)
Using Gage’s inequality (1.2) and the classical isoperimetric inequality (1.1), one finds that for a closed convex plane curve Z
k 2 ds ≥
α
πL 4π 2 ≥ . A L
(3.2)
(3.2) implies that Gage’s inequality is sharper than the Cauchy-Schwartz inequality for convex closed plane curves. As we know, Gage ([3]) has completed the proof of his inequality by using deep geometric considerations. Now, using (2.4) – (2.8), one can restate Gage’s inequality in the following manner which can be thought of as a mathematical analysis version of the inequality: Z
2π
00
p(θ)[p(θ) + p (θ)]dθ
0
Z
2π
0
Z 2π dθ ≥ 2π p(θ)dθ, p(θ) + p00 (θ) 0
where p(θ) is a periodic function with period 2π and satisfying p(θ) + p00 (θ) > 0.
p(θ) > 0,
As an isoperimetric-type inequality, one should require that the equality holds if and only if the curve is a circle, but Gage does not do that. Therefore, one can restate Gage’s inequality and strengthen it as the following more isoperimetric form Conjecture. Let p(θ) be a C 2 periodic function with period 2π and p(θ) > 0 and p(θ) + p00 (θ) > 0. Then Z
2π
p(θ)[p(θ) + p00 (θ)]dθ
0
Z 0
2π
Z 2π dθ ≥ 2π p(θ)dθ, p(θ) + p00 (θ) 0
(3.3)
And moreover, the equality holds if and only if p(θ) + p00 (θ) = c, where c is a positive constant.
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(3.4)
The statement of this conjecture is motivated by one of the proofs of the classical isoperimetric inequality (1.1) given by Hurwitz [11] in terms of the Fourier series of the support function p(θ). It seems to us that it is very difficult to answer this conjecture. Solving (3.4) yields that p(θ) = a cos θ + b sin θ + c, which implies that the convex curve with p(θ) as its Minkowski support function is a circle with radius c. Thus, Gage’s inequality holds for circles, what about the inverse? In all, we are expecting a mathematical analysis proof of this conjecture. Observing that inequality (3.1) can also be written as an analytic inequality Z 0
2π
Z 2π dθ p(θ)dθ ≥ 4π 2 , p(θ) + p00 (θ) 0
with equality when and only when p(θ) + p00 (θ) = c (a positive constant). This can R R easily be seen from the fact that 02π p(θ)dθ = 02π [p(θ) + p00 (θ)]dθ and the CauchySchwartz inequality.
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