Skip to content

Advertisement

  • Research
  • Open Access

Bonnesen-style inequalities on surfaces of constant curvature

Journal of Inequalities and Applications20182018:325

https://doi.org/10.1186/s13660-018-1918-1

  • Received: 17 October 2018
  • Accepted: 14 November 2018
  • Published:

Abstract

In this paper, some Bonnesen-style inequalities on a surface \(\mathbb {X}_{\kappa}\) of constant curvature κ (i.e., the Euclidean plane \(\mathbb{R}^{2}\), projective plane \(\mathbb{R}P^{2}\), or hyperbolic plane \(\mathbb{H}^{2}\)) are proved. The method is integral geometric and gives a uniform proof of some Bonnesen-style inequalities alone with equality conditions.

Keywords

  • Isoperimetric inequality
  • Bonnesen-style inequality
  • Isoperimetric deficit
  • Surface of constant curvature

MSC

  • 52A22

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, 2934]. 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, 2326]
$$ P_{K}^{2}-(4\pi-\kappa A_{K})A_{K} \geq0, $$
(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, 59, 11, 14, 17, 19, 2326, 2934].

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].

Lemma 2.1

Let K, L be two compact sets bounded by rectifiable simple closed curves in \(\mathbb {X}_{\kappa}\), then
$$ \int_{\mu}dg \geq2\pi(A_{K}+A_{L})-P_{K}P_{L}- \kappa A_{K}A_{L}. $$
(2.4)

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].

Lemma 2.2

For \(K\in\mathcal{C}(\mathbb{X}_{\kappa})\), let \(R_{K}\) and \(r_{K}\) be respectively the maximum inscribed radius and the minimum circumscribed radius of K. Then, for \(r_{K}\leq r \leq R_{K}\),
$$ \bigl[ (2\pi-\kappa A_{K} )^{2}+\kappa P_{K}^{2} \bigr] \operatorname{sn}^{2}_{\kappa}(r) -4\pi P_{K} \operatorname{sn}_{\kappa}(r) -A_{K}( \kappa A_{K}-4\pi) \leq0. $$
(2.5)

Proof

Let L be a geodesic disc of radius r. Then neither \(gB_{\kappa}(r)\subset K\) nor \(gB_{\kappa}(r)\supset K\) for any \(g\in G_{\kappa}\) and hence the measure \(\int_{\mu}dg =0\). By (1.5) and (2.4) we have
$$ P_{K} \operatorname{sn}_{\kappa}(r) - \biggl( \frac{2\pi}{\kappa}-A_{K} \biggr) \bigl(1-\operatorname{cn}_{\kappa}(r) \bigr)-A_{K} \geq0. $$
(2.6)
Identity (1.9) shows \(1-{\kappa}\cdot\operatorname{sn}^{2}_{\kappa}(r)= \operatorname{cn}^{2}_{\kappa}(r)>0\) and inequality (2.6) can be rewritten as
$$ P_{K} \operatorname{sn}_{\kappa}(r)- \frac{2\pi}{\kappa} \geq \biggl(A_{K}-\frac{2\pi}{\kappa} \biggr) \sqrt {1-{\kappa}\cdot\operatorname{sn}^{2}_{\kappa}(r)}. $$
(2.7)
For \(\kappa\geq0\),
$$ P_{K} \operatorname{sn}_{\kappa}(r)- \frac{2\pi}{\kappa}\leq0, $$
hence by squaring both sides of (2.7) we have
$$ \biggl(P_{K} \operatorname{sn}_{\kappa}(r)-\frac{2\pi}{\kappa} \biggr)^{2} \leq \biggl(A_{K}-\frac{2\pi}{\kappa} \biggr)^{2} \bigl(1-{\kappa}\cdot\operatorname{sn}^{2}_{\kappa}(r) \bigr), $$
that is,
$$ \bigl( (2\pi-\kappa A_{K} )^{2}+\kappa P_{K}^{2} \bigr) \operatorname{sn}^{2}_{\kappa}(r) -4\pi P_{K} \operatorname{sn}_{\kappa}(r) -A_{K}(\kappa A_{K}-4 \pi) \leq 0. $$
For \(\kappa<0\), then \(A_{K}-\frac{2\pi}{\kappa}>0\). Squaring both sides of (2.7) leads to
$$ \biggl(P_{K} \operatorname{sn}_{\kappa}(r)-\frac{2\pi}{\kappa} \biggr)^{2} \geq \biggl(A_{K}-\frac{2\pi}{\kappa} \biggr)^{2} \bigl(1-{\kappa}\cdot\operatorname{sn}^{2}_{\kappa}(r) \bigr), $$
i.e.,
$$ \bigl( (2\pi-\kappa A_{K} )^{2}+\kappa P_{K}^{2} \bigr) \operatorname{sn}^{2}_{\kappa}(r) -4\pi P_{K} \operatorname{sn}_{\kappa}(r) -A_{K}(\kappa A_{K}-4 \pi) \leq0. $$
 □

We are now in the position to prove our Bonnesen-style inequalities.

Theorem 2.1

Let \(K\in\mathcal{C}(\mathbb{X}_{\kappa})\). If \(r_{K}\leq r \leq R_{K}\), then
$$ \Delta_{\kappa}(K) \geq \frac{ (P_{K}-2\pi\operatorname{sn}_{\kappa}(r) )^{2}}{ \operatorname{cn}^{2}_{\kappa}(r)}, $$
(2.8)
with equality if K is a geodesic disc.

Proof

Inequality (2.5) can be rewritten as
$$ P_{K}^{2}-A_{K}(4\pi-\kappa A_{K}) \geq P_{K}^{2} + \bigl( (2\pi-\kappa A_{K} )^{2}+\kappa P_{K}^{2} \bigr) \operatorname{sn}^{2}_{\kappa}(r) - 4\pi P_{K} \operatorname{sn}_{\kappa}(r). $$
Since \((2\pi-\kappa A_{K} )^{2}+\kappa P_{K}^{2}=4\pi^{2}+\kappa\Delta_{\kappa}(K)\), we have
$$ \bigl(1-\kappa\ \operatorname{sn}^{2}_{\kappa}(r) \bigr) \Delta_{\kappa}(K) \geq \bigl(P_{K}-2\pi\operatorname{sn}_{\kappa}(r) \bigr)^{2}. $$
Via (1.9), that is, \(1-{\kappa}\cdot\operatorname{sn}^{2}_{\kappa}(r)=\operatorname {cn}^{2}_{\kappa}(r)\), then the previous inequality results in (2.8) and we complete the proof. □

Theorem 2.2

Let \(K\in\mathcal{C}(\mathbb{X}_{\kappa})\), then
$$ \begin{aligned}[b]\Delta_{\kappa}(K) \geq{}& c \bigl\{ 2\pi^{2} \bigl[\operatorname{sn}_{\kappa}(R_{K}) -\operatorname{sn}_{\kappa}(r_{K}) \bigr]^{2} +2 \bigl[P_{K}-\pi\operatorname{sn}_{\kappa}(R_{K}) -\pi\operatorname{sn}_{\kappa}(r_{K}) \bigr]^{2} \\ & -\kappa \bigl[\operatorname{sn}^{2}_{\kappa}(R_{K}) \bigl(P_{K}-2\pi\operatorname{sn}_{\kappa}(r_{K}) \bigr)^{2} + \operatorname{sn}^{2}_{\kappa}(r_{K}) \bigl(P_{K}-2\pi\operatorname{sn}_{\kappa}(R_{K}) \bigr)^{2} \bigr] \bigr\} , \end{aligned} $$
where \(c=1/({2\operatorname{cn}^{2}_{\kappa}(R_{K}) \operatorname{cn}^{2}_{\kappa}(r_{K})})\), and the equality holds if K is a geodesic disc.

Proof

Inequality (2.8) holds for \(r=R_{K}\) and \(r=r_{K}\), respectively:
$$\begin{gathered} \Delta_{\kappa}(K) \geq\frac{ (P_{K}-2\pi\operatorname{sn}_{\kappa}(R_{K}) )^{2}}{ \operatorname{cn}^{2}_{\kappa}(R_{K})}; \\ \Delta_{\kappa}(K) \geq \frac{ (P_{K}-2\pi\operatorname{sn}_{\kappa}(r_{K}) )^{2}}{ \operatorname{cn}^{2}_{\kappa}(r_{K})}.\end{gathered} $$
By adding the above two inequalities side by side, we have
$$\begin{aligned} \Delta_{\kappa}(K) \geq{}& c \bigl[\operatorname{cn}^{2}_{\kappa}(r_{K}) \bigl(P_{K}-2\pi\operatorname{sn}_{\kappa}(R_{K}) \bigr)^{2} +\operatorname{cn}^{2}_{\kappa}(R_{K}) \bigl(P_{K}-2\pi\operatorname{sn}_{\kappa}(r_{K}) \bigr)^{2} \bigr] \\ ={}&c \bigl[\bigl(1-\kappa\operatorname{sn}^{2}_{\kappa}(r_{K}) \bigr) \bigl(P_{K}-2\pi\operatorname{sn}_{\kappa}(R_{K}) \bigr)^{2} +\bigl(1-\kappa\operatorname{sn}^{2}_{\kappa}(R_{K}) \bigr) \bigl(P_{K}-2\pi\operatorname{sn}_{\kappa}(r_{K}) \bigr)^{2} \bigr] \\ ={}&c \bigl\{ \bigl(P_{K}-2\pi\operatorname{sn}_{\kappa}(R_{K}) \bigr)^{2} + \bigl(P_{K}-2\pi\operatorname{sn}_{\kappa}(r_{K}) \bigr)^{2} \\ & -\kappa\bigl[\operatorname{sn}^{2}_{\kappa}(R_{K}) \bigl(P_{K}-2\pi\operatorname{sn}_{\kappa}(r_{K}) \bigr)^{2} + \operatorname{sn}^{2}_{\kappa}(r_{K}) \bigl(P_{K}-2\pi\operatorname{sn}_{\kappa}(R_{K}) \bigr)^{2} \bigr] \bigr\} .\end{aligned} $$
Via elementary calculations we obtain the desired Bonnesen-style inequality. □

For \(\kappa<0\), the following Bonnesen-style inequalities are immediate consequences of Theorem 2.2 with equality conditions.

Corollary 2.1

Let \(K\in\mathcal{C}(\mathbb{H}^{2})\), then
$$ \Delta_{\kappa}(K) \geq 2c \bigl\{ \pi^{2} \bigl[\operatorname{sn}_{\kappa}(R_{K}) -\operatorname{sn}_{\kappa}(r_{K}) \bigr]^{2} + \bigl[P_{K}-\pi\operatorname{sn}_{\kappa}(R_{K}) -\pi\operatorname{sn}_{\kappa}(r_{K}) \bigr]^{2} \bigr\} , $$
(2.9)
where \(c=1/({2\operatorname{cn}^{2}_{\kappa}(R_{K}) \operatorname{cn}^{2}_{\kappa}(r_{K})})\), and the equality holds if K is a hyperbolic disc.

Corollary 2.2

Let \(K\in\mathcal{C}(\mathbb{H}^{2})\), then
$$ \Delta_{\kappa}(K) \geq 2c \bigl(P_{K}-\pi \operatorname{sn}_{\kappa}(R_{K}) -\pi\operatorname{sn}_{\kappa}(r_{K}) \bigr)^{2}, $$
(2.10)
where \(c={1}/({2\operatorname{cn}^{2}_{\kappa}(R_{K}) \operatorname{cn}^{2}_{\kappa}(r_{K})})\), with equality if and only if K is a hyperbolic disc.

Proof

Since
$$ \begin{gathered}\pi^{2} \bigl[\operatorname{sn}_{\kappa}(R_{K}) -\operatorname{sn}_{\kappa}(r_{K}) \bigr]^{2} + \bigl[P_{K}-\pi\operatorname{sn}_{\kappa}(R_{K}) -\pi \operatorname{sn}_{\kappa}(r_{K}) \bigr]^{2} \\ \quad\geq \bigl[P_{K}-\pi\operatorname{sn}_{\kappa}(R_{K}) -\pi\operatorname{sn}_{\kappa}(r_{K}) \bigr]^{2},\end{gathered} $$
with equality holds if and only if \(R_{K} =r_{K}\), that is, K must be a hyperbolic disc, the Bonnesen-style inequality (2.10) follows from inequality (2.9) immediately. □

3 Bonnesen-style inequalities in \(\mathbb{H}^{2}\)

We are seeking more Bonnesen-style inequalities in \(\mathbb{H}^{2}\).

Theorem 3.1

Let \(K\in\mathcal{C}(\mathbb{H}^{2})\), then
$$ \Delta_{\kappa}(K) \geq \pi^{2} \bigl( \operatorname{tn}_{\kappa}(R_{K})-\operatorname{tn}_{\kappa }(r_{K}) \bigr)^{2} + \frac{P_{K}^{2}}{4} \biggl(\frac{1}{\operatorname{cn}_{\kappa}(r_{K})} - \frac{1}{\operatorname{cn}_{\kappa}(R_{K})} \biggr)^{2}, $$
(3.1)
with equality if K is a hyperbolic disc.

Proof

By Theorem 2.1, \(r=R_{K}\) and \(r=r_{K}\) respectively lead to
$$ \Delta_{\kappa}(K) \geq \frac{ (2\pi\operatorname{sn}_{\kappa}(R_{K})-P_{K} )^{2}}{ \operatorname{cn}^{2}_{\kappa}(R_{K})} \quad\text{and} \quad\Delta _{\kappa}(K) \geq \frac{ (P_{K}-2\pi\operatorname{sn}_{\kappa}(r_{K}) )^{2}}{ \operatorname{cn}^{2}_{\kappa}(r_{K})}. $$
Adding two inequalities side by side and by the inequality \(x^{2}+y^{2}\geq\frac{(x+y)^{2}}{2}\), we have
$$\begin{aligned} 2\Delta_{\kappa}(K) &\geq \frac{ (2\pi\operatorname{sn}_{\kappa}(R_{K})-P_{K} )^{2}}{ \operatorname{cn}^{2}_{\kappa}(R_{K})}+ \frac{ (P_{K}-2\pi\operatorname{sn}_{\kappa}(r_{K}) )^{2}}{ \operatorname{cn}^{2}_{\kappa}(r_{K})} \\ &\geq\frac{1}{2} \biggl\{ 2\pi \bigl(\operatorname{tn}_{\kappa}(R_{K}) -\operatorname{tn}_{\kappa}(r_{K}) \bigr)+ \biggl( \frac {P_{K}}{\operatorname{cn}_{\kappa}(r_{K})} -\frac{P_{K}}{\operatorname{cn}_{\kappa}(R_{K})} \biggr) \biggr\} ^{2}.\end{aligned} $$
For \(K\in\mathcal{C}(\mathbb{H}^{2})\), \(\operatorname{tn}_{\kappa}\) and \(\operatorname{sn}_{\kappa}\) are respectively hyperbolic tangent \(\operatorname{tanh}(x)\) and hyperbolic cosine \(\operatorname{cosh}(x)\) that are strictly increasing on \([0,\infty)\). Therefore, for \(r_{K}\leq R_{K}\),
$$ \operatorname{tn}_{\kappa}(R_{K})-\operatorname{tn}_{\kappa }(r_{K}) \geq 0;\qquad \frac{1}{\operatorname{cn}_{\kappa}(r_{K})}- \frac{1}{\operatorname{cn}_{\kappa}(R_{K})}\geq0. $$
By inequality \((x+y)^{2}\geq x^{2}+y^{2}\) (\(x\geq0\), \(y\geq0\)), we have
$$ 2\Delta_{\kappa}(K) \geq2\pi^{2} \bigl(\operatorname{tn}_{\kappa}(R_{K}) -\operatorname{tn}_{\kappa}(r_{K}) \bigr)^{2}+ \frac{P_{K}^{2}}{2} \biggl(\frac{1}{\operatorname{cn}_{\kappa}(r_{K})} -\frac{1}{\operatorname{cn}_{\kappa}(R_{K})} \biggr)^{2}. $$
 □

The following Bonnesen-style inequality with equality condition for \(K\in\mathcal{C}(\mathbb{H}^{2})\) is a direct consequence of Theorem 3.1.

Corollary 3.1

Let \(K\in\mathcal{C}(\mathbb{H}^{2})\), then
$$ \Delta_{\kappa}(K) \geq \pi^{2} \bigl( \operatorname{tn}_{\kappa}(R_{K})-\operatorname{tn}_{\kappa }(r_{K}) \bigr)^{2}, $$
(3.2)
with equality if and only if K is a hyperbolic disc.

Proof

By slightly complicated elementary calculations, we have
$$ \pi^{2} \bigl(\operatorname{tn}_{\kappa}(R_{K})- \operatorname{tn}_{\kappa }(r_{K}) \bigr)^{2} + \frac{P_{K}^{2}}{4} \biggl(\frac{1}{\operatorname{cn}_{\kappa}(r_{K})} -\frac{1}{\operatorname{cn}_{\kappa}(R_{K})} \biggr)^{2}\geq \pi^{2} \bigl(\operatorname{tn}_{\kappa}(R_{K})- \operatorname{tn}_{\kappa }(r_{K}) \bigr)^{2}. $$
(3.3)
Then inequality (3.2) follows from (3.1) and (3.3) immediately.

Equality holds in (3.2) and (3.3) if and only if either \(P_{K}=0\), which implies that K is a single point, or \(R_{K}=r_{K}\), which means that K is a hyperbolic disc. □

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.

Corollary 3.2

Let \(K\in\mathcal{C}(\mathbb{H}^{2})\), then
$$ \Delta_{\kappa}(K) \geq \frac{P_{K}^{2}}{4} \biggl( \frac{1}{\operatorname{cn}_{\kappa}(r_{K})} -\frac{1}{\operatorname{cn}_{\kappa}(R_{K})} \biggr)^{2}, $$
(3.4)
with equality if and only if K is a hyperbolic disc.

4 The limiting cases on the Euclidean plane \(\mathbb{R}^{2}\)

The limiting cases of Bonnesen-style inequalities obtained in the previous sections are known as Bonnesen-style inequalities in the Euclidean plane \(\mathbb{R}^{2}\).

Corollary 4.1

Let K be a compact convex set in \(\mathbb{R}^{2}\). If \(r_{K}\leq r \leq R_{K}\), then
$$ P_{K}^{2}-4\pi A_{K}\geq (P_{K}-2\pi r )^{2}, $$
(4.1)
with equality if K is a Euclidean disc.

Proof

For \(\kappa<0\), let \(\kappa=-\frac{1}{R^{2}}\). Then inequality (2.8) becomes
$$ P_{K}^{2}-4\pi A_{K}-\frac{A_{K}^{2}}{R^{2}} \geq \frac{ (P_{K}-2\pi R\sinh(\frac{r}{R}) )^{2}}{\cosh^{2}(\frac{r}{R})}. $$
As \(R\rightarrow\infty\), the inequality above leads to the following inequality by L’Hôpital’s rule:
$$ P_{K}^{2}-4\pi A_{K} \geq \lim _{R\rightarrow\infty} \biggl(P_{K}-2\pi R\sinh \biggl( \frac{r}{R} \biggr) \biggr)^{2} = (P_{K}-2\pi r )^{2}. $$
 □

Corollary 4.2

Let K be a compact convex set in \(\mathbb{R}^{2}\), then
$$ P_{K}^{2}-4\pi A_{K} \geq \pi^{2} (R_{K}-r_{K} )^{2}, $$
(4.2)
with equality if and only if K is a Euclidean disc.

Proof

For \(\kappa=-\frac{1}{R^{2}}\), inequality (3.2) becomes
$$ P_{K}^{2}-4\pi A_{K}-\frac{A_{K}^{2}}{R^{2}} \geq \pi^{2} \biggl(\frac{R\sinh(\frac{R_{K}}{R})}{\cosh(\frac{R_{K}}{R})} -\frac{R\sinh(\frac{r_{K}}{R})}{\cosh(\frac{r_{K}}{R})} \biggr)^{2}. $$
As \(R\rightarrow\infty\), the inequality above becomes
$$ P_{K}^{2}-4\pi A_{K} \geq\lim _{R\rightarrow\infty} \biggl(R\sinh \biggl(\frac{R_{K}}{R} \biggr)-R\sinh \biggl(\frac {r_{K}}{R} \biggr) \biggr)^{2} = \pi^{2} (R_{K}-r_{K} )^{2}. $$
The inequality holds as an equality if and only if \(R_{K}=r_{K}\), that is, K is a Euclidean disc. □

The limiting case of Theorem 2.2 is the following strengthening inequality of (4.2).

Corollary 4.3

Let K be a compact convex set in \(\mathbb{R}^{2}\). Then
$$ P_{K}^{2}-4\pi A_{K} \geq \pi^{2} (R_{K}-r_{K} )^{2} + (P_{K}-\pi R_{K}-\pi r_{K} )^{2}, $$
with equality if K is a Euclidean disc.

Declarations

Acknowledgements

The author would like to thank anonymous referees for encouraging and critical comments and suggestions that definitely led to improvements of the original manuscript.

Funding

The author was supported in part by the Fundamental Research Funds for the Central Universities (No. XDJK2016C167).

Authors’ contributions

The author read and approved the final manuscript.

Competing interests

The author declares that they have no competing interests.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Mathematics and Statistics, Southwest University, Chongqing, People’s Republic of China

References

  1. Banchoff, T., Pohl, W.: A generalization of the isoperimetric inequality. J. Differ. Geom. 6, 175–213 (1971) MathSciNetView ArticleGoogle Scholar
  2. Bokowski, J., Heil, E.: Integral representation of quermassintegrals and Bonnesen-style inequalities. Arch. Math. 47, 79–89 (1986) MathSciNetView ArticleGoogle Scholar
  3. Bonnesen, T.: Les probléms des isopérimétres et des isépiphanes. Gauthier-Villars, Paris (1929) MATHGoogle Scholar
  4. Burago, Yu.D., Zalgaller, V.A.: Geometric Inequalities. Springer, Berlin (1988) View ArticleGoogle Scholar
  5. Campi, S.: Three-dimensional Bonnesen type inequalities. Matematiche 60, 425–431 (2005) MathSciNetMATHGoogle Scholar
  6. Diskant, V.: A generalization of Bonnesen’s inequalities. Sov. Math. Dokl. 14, 1728–1731 (1973). Transl. of Dokl. Akad. Nauk SSSR 213 (1973) MathSciNetMATHGoogle Scholar
  7. Enomoto, K.: A generalization of the isoperimetric inequality on \(S^{2}\) and flat tori in \(S^{3}\). Proc. Am. Math. Soc. 120(2), 553–558 (1994) MATHGoogle Scholar
  8. Green, M., Osher, S.: Steiner polynomials, Wulff flows, and some new isoperimetric inequalities for convex plane curves. Asian J. Math. 3(3), 659–676 (1999) MathSciNetView ArticleGoogle Scholar
  9. Hsiung, C.C.: Isoperimetric inequalities for two-dimensional Riemannian manifolds with boundary. Ann. Math. 73(2), 213–220 (1961) MathSciNetView ArticleGoogle Scholar
  10. Hurwitz, A.: Sur quelques applications géométriques des séries de Fourier. Ann. Sci. Éc. Norm. Supér. 19, 358–408 (1902) MATHGoogle Scholar
  11. Klain, D.: An error estimate for the isoperimetric deficit. Ill. J. Math. 49(3), 981–992 (2005) MathSciNetMATHGoogle Scholar
  12. Klain, D.: Bonnesen-type inequalities for surfaces of constant curvature. Adv. Appl. Math. 39(2), 143–154 (2007) MathSciNetView ArticleGoogle Scholar
  13. Klain, D., Rota, G.: Introduction to Geometric Probability. Lezioni Lincee. Cambridge University Press, Cambridge (1997) MATHGoogle Scholar
  14. Ku, H., Ku, M., Zhang, X.: Isoperimetric inequalities on surfaces of constant curvature. Can. J. Math. 49, 1162–1187 (1997) MathSciNetView ArticleGoogle Scholar
  15. Osserman, R.: The isoperimetric inequality. Bull. Am. Math. Soc. 84, 1182–1238 (1978) MathSciNetView ArticleGoogle Scholar
  16. Osserman, R.: Bonnesen-style isoperimetric inequality. Am. Math. Mon. 86, 1–29 (1979) View ArticleGoogle Scholar
  17. Santaló, L.A.: Integral formulas in Crofton’s style on the sphere and some inequalities referring to spherical curves. Duke Math. J. 9, 707–722 (1942) MathSciNetView ArticleGoogle Scholar
  18. Santaló, L.A.: Integral geometry on surfaces of constant negative curvature. Duke Math. J. 10, 687–704 (1943) MathSciNetView ArticleGoogle Scholar
  19. Santaló, L.A.: Integral Geometry and Geometric Probability. Addison-Wesley, Reading (1976) MATHGoogle Scholar
  20. Schmidt, E.: Über das isoperimetrische problem im Raum von n dimensionen. Math. Z. 44, 689–788 (1939) MathSciNetView ArticleGoogle Scholar
  21. Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory. Cambridge University Press, Cambridge (2014) MATHGoogle Scholar
  22. Steiner, J.: Einfacher Beweis der isoperimetrischen Hauptsätze. J. Reine Angew. Math. 18, 281–296 (1838) MathSciNetView ArticleGoogle Scholar
  23. Teufel, E.: A generalization of the isoperimetric inequality in the hyperbolic plane. Arch. Math. 57(5), 508–513 (1991) MathSciNetView ArticleGoogle Scholar
  24. Teufel, E.: Isoperimetric inequalities for closed curves in spaces of constant curvature. Results Math. 22, 622–630 (1992) MathSciNetView ArticleGoogle Scholar
  25. Wei, S., Zhu, M.: Sharp isoperimetric inequalities and sphere theorems. Pac. J. Math. 220(1), 183–195 (2005) MathSciNetView ArticleGoogle Scholar
  26. Weiner, J.: A generalization of the isoperimetric inequality on the 2-sphere. Indiana Univ. Math. J. 24, 243–248 (1974) MathSciNetView ArticleGoogle Scholar
  27. Xu, W., Zhou, J., Zhu, B.: On Bonnesen-type inequalities for a surface of constant curvature. Proc. Am. Math. Soc. 143, 4925–4935 (2015) MathSciNetView ArticleGoogle Scholar
  28. Zeng, C., Ma, L., Zhou, J., Chen, F.: The Bonnesen isoperimetric inequality in a surface of constant curvature. Sci. China Math. 55(9), 1913–1919 (2012) MathSciNetView ArticleGoogle Scholar
  29. Zeng, C., Zhou, J., Yue, S.: The symmetric mixed isoperimetric inequality of two planar convex domains. Acta Math. Sin. 55(2), 355–362 (2012) MathSciNetMATHGoogle Scholar
  30. Zhou, J.: On Bonnesen-type inequalities. Acta Math. Sin. (Chin. Ser.) 50(6), 1397–1402 (2007) MathSciNetMATHGoogle Scholar
  31. Zhou, J., Chen, F.: The Bonnesen-type inequality in a plane of constant curvature. J. Korean Math. Soc. 44(6), 1363–1372 (2007) MathSciNetView ArticleGoogle Scholar
  32. Zhou, J., Du, Y., Cheng, F.: Some Bonnesen-style inequalities for higher dimensions. Acta Math. Sin. 28(12), 2561–2568 (2012) MathSciNetView ArticleGoogle Scholar
  33. Zhou, J., Ren, D.: Geometric inequalities from the viewpoint of integral geometry. Acta Math. Sci. Ser. A (Chin. Ed.) 30(5), 1322–1339 (2010) MathSciNetMATHGoogle Scholar
  34. Zhou, J., Xia, Y., Zeng, C.: Some new Bonnesen-style inequalities. J. Korean Math. Soc. 48, 421–430 (2011) MathSciNetView ArticleGoogle Scholar

Copyright

© The Author(s) 2018

Advertisement