1 Introduction
The classical isoperimetric problem dates back to antique literature and geometry. The problem can be stated as: Among all closed curves of given length in the Euclidean plane \(\mathbb {R}^{2}\), which one maximizes the area of its enclosed region?
The solution to the problem is usually expressed in the form of an inequality that relates the length
\(P_{K}\) of a rectifiable simple closed curve and the area
\(A_{K}\) of the planar region
K that the curve encloses in
\(\mathbb{R}^{2}\). The solution to the classical isoperimetric problem is characterized as the following isoperimetric inequality:
$$ P_{K}^{2}-4\pi A_{K}\geq0, $$
(1.1)
with equality if and only if
K is a Euclidean disc.
The history of geometric proofs for the classical isoperimetric problem goes back to Ancient Greeks and was recorded by Pappus of Alexandria in the fourth century AD, but their arguments were incomplete. The first progress towards the solution was made by Steiner [
22] in 1838 by a geometric method later named Steiner symmetrization. His proof contained a flaw that later was fixed by analytic approach. In 1870, Weierstrass gave the first rigorous proof as a corollary of his theory of calculus of several variables. Since then, many other proofs have been discovered. In 1902, Hurwitz [
10] published a short proof using the Fourier series that applies to arbitrary rectifiable curves (not assumed to be smooth). An elegant direct proof based on comparison of a simple closed curve of length
L with the circle of radius
\(\frac{L}{2\pi}\) was given by Schmidt [
20]. See [
4,
15,
16,
21] for more references.
In 1920s, Bonnesen proved a series of inequalities of the form [
3]
$$ P_{K}^{2}-4\pi A_{K}\geq B_{K}, $$
(1.2)
where
\(B_{K}\) is a non-negative invariant and vanishes if and only if the domain
K is a Euclidean disc.
A well-known Bonnesen-style inequality is
$$ P_{K}^{2}-4\pi A_{K}\geq \pi^{2}(R_{{K}}-r_{K})^{{2}}, $$
(1.3)
where
\(R_{K}\) and
\(r_{K}\) respectively denote the circumradius and inradius of
K, with equality if and only if
K is a Euclidean disc.
Many inequalities of style (
1.2), called Bonnesen-style inequalities, were found along with variations and generalizations in the past decades [
2,
5,
6,
8,
11,
19,
29‐
34]. On the other hand, the classical isoperimetric inequality has been extended to higher dimensions and a surface of constant curvature
κ, i.e., the Euclidean plane
\(\mathbb{R}^{2}\), projective plane
\(\mathbb{R}P^{2}\), or hyperbolic plane
\(\mathbb{H}^{2}\).
Let
K be a compact set bounded by a rectifiable simple closed curve with the area
\(A_{K}\) and perimeter
\(P_{K}\) in
\(\mathbb{X}_{\kappa}\). Then [
1,
7,
9,
14,
17,
18,
23‐
26]
$$ P_{K}^{2}-(4\pi-\kappa A_{K})A_{K} \geq0, $$
(1.4)
with equality if and only if
K is a geodesic disc.
The geodesic disc of radius
r with center
x is defined as
$$B_{\kappa}(x, r)=\bigl\{ y\in\mathbb{X}_{\kappa}: d(x,y)\leq r\bigr\} , $$
where
d is the geodesic distance function in
\(\mathbb{X}_{\kappa}\). The area, perimeter of
\(B_{\kappa}(x,r)\) in
\(\mathbb{X}_{\kappa}\) are respectively [
12]
$$ A\bigl(B_{\kappa}(x,r)\bigr)=\frac{2\pi}{\kappa}\bigl(1- \operatorname{cn}_{\kappa}(r)\bigr), \qquad P\bigl(B_{\kappa}(x,r) \bigr)=2\pi\operatorname{sn}_{\kappa}(r). $$
(1.5)
The limiting cases of as
\(\kappa\rightarrow0\) yield the Euclidean formulas
\(A(B(x,r))=\pi r^{2}\) and
\(P(B(x,r))=2\pi r\).
A Bonnesen-type inequality in
\(\mathbb{X}_{\kappa}\) is of the form
$$ P^{2}_{K}-(4\pi-\kappa A_{K})A_{K} \geq B_{K}, $$
(1.6)
where
\(B_{K}\) vanishes if and only if
K is a geodesic disc [
15,
28].
Bonnesen [
3] established an inequality of the type (
1.6) in the sphere of radius
\(1/\sqrt{\kappa}\):
$$ P^{2}_{K}-(4\pi-\kappa A_{K})A_{K} \geq4\pi^{2}\ \operatorname{tn}_{\kappa}^{2} \biggl( \frac{R_{K}-r_{K}}{2} \biggr), $$
(1.7)
where
\(R_{K}\) and
\(r_{K}\) are respectively the minimum circumscribed radius and the maximum inscribed radius of
K.
Let
\(\mathbb{X}_{\kappa}\) be the surface of constant curvature
κ, specifically:
$$ \mathbb{X}_{\kappa}= \textstyle\begin{cases} \mathbb{P}R^{2}, \text{Euclidean 2-sphere of radius } 1/\sqrt{\kappa}, & \text{if } \kappa>0;\\ \mathbb{R}^{2}, \text{Euclidean plane}, &\text{if } \kappa=0;\\ \mathbb{H}^{2}, \text{Hyperbolic plane of constant curvature } \kappa, & \text{if } \kappa< 0. \end{cases} $$
Let
$$ \Delta_{\kappa}(K) =P^{2}_{K}-(4\pi- \kappa A_{K})A_{K} $$
(1.8)
denote the isoperimetric deficit of
K in
\(\mathbb{X}_{\kappa}\). The trigonometric functions appearing in (
1.7) are defined by
$$\begin{gathered} \operatorname{sn}_{\kappa}(t) = \textstyle\begin{cases} \frac{1}{\sqrt{-\kappa}} \sinh(\sqrt{-\kappa} t), &\kappa< 0,\\ t, & \kappa=0,\\ \frac{1}{\sqrt{\kappa}} \sin(\sqrt{\kappa} t),& \kappa>0; \end{cases}\displaystyle \\ \operatorname{cn}_{\kappa}(t)= \textstyle\begin{cases} \cosh(\sqrt{-\kappa}t), &\kappa< 0,\\ 1, &\kappa=0,\\ \cos(\sqrt{\kappa}t), &\kappa>0; \end{cases}\displaystyle \\\operatorname{tn}_{\kappa}(t)= \frac{\operatorname{sn}_{\kappa}(t)}{\operatorname{cn}_{\kappa}(t)}; \qquad \operatorname{ct}_{\kappa}(t)= \frac{\operatorname{cn}_{\kappa}(t)}{\operatorname{sn}_{\kappa}(t)};\end{gathered} $$
and
$$ \textstyle\begin{array}{lll} {\kappa}\cdot\operatorname{sn}^{2}_{\kappa}(t)+ \operatorname{cn}^{2}_{\kappa}(t)=1. \end{array} $$
(1.9)
The following Bonnesen-type inequality is obtained in [
31]:
$$ \Delta_{\kappa}(K) \geq \biggl(2\pi-\frac{\kappa}{2} A_{K} \biggr)^{2} \biggl(\operatorname{tn}_{\kappa} \frac{R_{K}}{2} -\operatorname{tn}_{\kappa}\frac{r_{K}}{2} \biggr)^{2} $$
(1.10)
for a convex set
K, with equality if
K is a geodesic disc.
Inequality (
1.10) was strengthened [
28] as
$$ \begin{aligned}[b] \Delta_{\kappa}(K) \geq{}& \biggl(2\pi-\frac{\kappa}{2} A_{K} \biggr)^{2} \biggl(\operatorname{tn}_{\kappa}\frac{R_{K}}{2} -\operatorname{tn}_{\kappa}\frac{r_{K}}{2} \biggr)^{2}\\ & + \biggl(2\pi-\frac{\kappa}{2} A_{K} \biggr)^{2} \biggl(\operatorname{tn}_{\kappa}\frac{R_{K}}{2} +\operatorname{tn}_{\kappa}\frac{r_{K}}{2} -\frac{2P_{K}}{4\pi-\kappa A_{K}} \biggr)^{2}, \end{aligned} $$
(1.11)
with equality if
K is a geodesic disc.
For a convex set
K in
\(\mathbb{X}_{\kappa}\) such that
\((2\pi-\kappa A_{K})^{2}+\kappa P_{K}^{2}\geq0\) if
\(\kappa<0\), Klain [
12] obtained the following Bonnesen-style inequality:
$$ \Delta_{\kappa}(K) \geq \frac{ ( (2\pi-\kappa A_{K} )^{2} +\kappa P_{K}^{2} )^{2}}{ 4(2\pi-\kappa A_{K})^{2}} \bigl( \operatorname{sn}_{\kappa}(R_{K})-\operatorname{sn}_{\kappa }(r_{K}) \bigr)^{2}, $$
(1.12)
with equality if
K is a geodesic disc.
For more results on Bonnesen-style inequality, see, e.g., [
1,
2,
5‐
9,
11,
14,
17,
19,
23‐
26,
29‐
34].
The purpose of this paper is to find a new Bonnesen-style inequality with equality condition on surfaces
\(\mathbb{X}_{\kappa}\) of constant curvature, especially on the hyperbolic plane
\(\mathbb{H}^{2}\) by integral geometric method. We are going to seek the following Bonnesen-style inequality for a convex set
K in
\(\mathbb{X}_{\kappa}\):
$$ \Delta_{\kappa}(K) \geq \pi^{2} \bigl(\operatorname{tn}_{\kappa}(R_{K})- \operatorname{tn}_{\kappa }(r_{K}) \bigr)^{2}, $$
with equality if and only if
K is a hyperbolic disc.
Finally, we give some special cases of these Bonnesen-style inequalities that strengthen some known Bonnesen-style inequalities in the Euclidean plane including the Bonnesen isoperimetric inequality (
1.3).
2 Preliminaries
Let \(\mathcal{C}(\mathbb{X}_{\kappa})\) be the set of all convex sets with perimeter \(P_{K}\leq \frac{2\pi}{\sqrt{\kappa}}\) if \(\kappa>0\) in \(\mathbb{X}_{\kappa}\). For a fixed point \(x_{0}\in\mathbb{X}_{\kappa}\), the geodesic disc of radius r with center \(x_{0}\) is the set of points that lie at most a distance r from \(x_{0}\) in \(\mathbb{X}_{\kappa}\). For \(K\in\mathbb{X}_{\kappa}\), let \(A_{K}\) and \(P_{K}\) denote the area and the perimeter of K, respectively. Let \(r_{K}\) and \(R_{K}\) be the maximum inscribed radius and the minimum circumscribed radius of K, respectively. We always assume that K lies in an open hemisphere of \(\mathbb {P}R^{2}\) such that \(R_{K}<\frac{\pi}{2\sqrt{\kappa}}\).
A set K is convex if, for points \(x, y\in K\), the shortest geodesic curve connecting x, y belongs to K. It should be noted that, for a convex set K in \(\mathbb{P}R^{2}\), \(2\pi-\kappa A_{K}>0\).
Let \(G_{\kappa}\) be the group of isometries in \(\mathbb {X}_{\kappa}\), and let dg be the kinematic density (Harr measure) on \(G_{\kappa}\). Let K be fixed and gL as moving via the isometry \(g\in G_{\kappa}\). For \(K, L\in\mathbb{X}_{\kappa}\), let \(\chi(K\cap gL)\) and \(\sharp(\partial K\cap \partial(gL))\) be the Euler–Poincaré characteristic of \(K\cap gL\) and the number of points of the intersection \(\partial K\cap\partial (gL)\), respectively.
The following fundamental kinematic formula is due to Blaschke [
19]:
$$ \int_{\{g: K\cap gL\ne\emptyset\}} \chi(K\cap gL) \,dg =2\pi(A_{K}+A_{L})+P_{K}P_{L}- \kappa A_{K}A_{L}. $$
(2.1)
As the limiting case, when
K,
L degenerate to curves
∂K,
∂L, respectively, then
\(A_{K}=A_{L}=0\) and the perimeters are
\(2P_{K}\),
\(2P_{L}\). Hence we have the following kinematic formula of Poincaré [
19]:
$$ \int_{\{g: \partial K\cap\partial(gL)\ne\emptyset\}} \sharp \bigl(\partial K\cap \partial(gL)\bigr) \,dg =4P_{K}P_{L}. $$
(2.2)
Since the compact sets are assumed to be simply connected and enclosed by simple curves,
\(\chi (K\cap gL)=n(g)\equiv (\text{the number of connected components of the intersection }K\cap gL)\). Let
\(\mu=\{g \in G_{\kappa}: K\subset gL \text{ or } K\supset gL \}\), then the fundamental kinematic formula of Blaschke (
2.1) can be rewritten as [
31]:
$$ \int_{\mu}dg + \int_{\{g: \partial K\cap\partial(gL)\ne\emptyset\}} n(g) \,dg =2\pi(A_{K}+A_{L})+P_{K}P_{L}- \kappa A_{K}A_{L}. $$
(2.3)
When
\(\partial K\cap\partial(gL)\ne\emptyset\), each component of
\(K\cap gL\) is bounded by at least an arc of
∂K and an arc of
\(\partial(g L)\), and
\(n(g)\le\sharp(\partial K\cap\partial(gL))/2\). Then the following containment measure inequality is an immediate consequence of Poincaré ’s formula (
2.2) and Blaschke’s formula (
2.3) [
12,
13,
19].
If
\(K\equiv L\), then there is no
\(g\in G_{\kappa}\) such that
\(gK\supset K\) nor
\(gK\subset K\). Hence
\(\int_{\mu}dg=0\) and inequality (
2.4) immediately leads to the isoperimetric inequality (
1.2).
The following Bonnesen inequality in
\(\mathbb {X}_{\kappa}\) is important for our main results [
27].
We are now in the position to prove our Bonnesen-style inequalities.
For
\(\kappa<0\), the following Bonnesen-style inequalities are immediate consequences of Theorem
2.2 with equality conditions.
3 Bonnesen-style inequalities in \(\mathbb{H}^{2}\)
We are seeking more Bonnesen-style inequalities in \(\mathbb{H}^{2}\).
The following Bonnesen-style inequality with equality condition for
\(K\in\mathcal{C}(\mathbb{H}^{2})\) is a direct consequence of Theorem
3.1.
Since
$$ \pi^{2} \bigl(\operatorname{tn}_{\kappa}(R_{K})- \operatorname{tn}_{\kappa }(r_{K}) \bigr)^{2} + \frac{P_{K}^{2}}{4} \biggl(\frac{1}{\operatorname{cn}_{\kappa}(r_{K})} -\frac{1}{\operatorname{cn}_{\kappa}(R_{K})} \biggr)^{2}\geq \frac{P_{K}^{2}}{4} \biggl(\frac{1}{\operatorname{cn}_{\kappa}(r_{K})} - \frac{1}{\operatorname{cn}_{\kappa}(R_{K})} \biggr)^{2}, $$
with equality if and only if
\(R_{K}=r_{K}\), which implies that
K must be a hyperbolic disc.
Combining this inequality with inequality (
3.1) immediately leads to the following Bonnesen-style inequality.
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