1 Introduction and main results
Integral geometry originated from geometric probability. It is a very important branch of the global differential geometry, which investigates the global properties of manifolds and convex bodies. Geometric inequality is an important topic in integral geometry. Perhaps the classical isoperimetric inequality is the oldest geometric inequality, that is, the disc encloses the maximum area among all domains of fixed perimeter. Let
K be a domain of area
A with simple boundary of perimeter
P in
\(\mathbb{R}^{2}\), then
$$ P^{2}-4\pi A\ge 0, $$
(1.1)
the equality sign holds if and only if
K is a disc.
The root of the classical isoperimetric problem can be traced back to ancient Greece. However, the rigorous mathematical proof of the isoperimetric inequality was obtained in the 19th century. Via the variational method, the first rigorous mathematical proof of the isoperimetric inequality was obtained by Weierstrass in 1870. By comparing a simple closed curve and a circle, Schmidt found a concise proof of the isoperimetric inequality in 1938. The isoperimetric inequality has been extended to the discrete case, the higher dimensions, and the surface of constant curvature (see [
1,
2,
6,
9‐
11,
15,
18,
22,
31‐
35]).
The quantity of the isoperimetric inequality (
1.1)
$$ \Delta _{2}(K)=P^{2}-4\pi A $$
(1.2)
measures the deficit between
K and a disc of radius
\(P/2\pi \), it is called the isoperimetric deficit of
K.
During the 1920s, Bonnesen proved some inequalities of the following form:
$$ \Delta _{2} (K)=P^{2}-4\pi A\ge B_{K}, $$
(1.3)
where
\(B_{K}\) is a nonnegative invariant of geometric significance and
\(B_{K}=0\) if and only if
K is a disc. An inequality of the form (
1.3) is called the Bonnesen-style inequality, and it is stronger than the isoperimetric inequality (
1.1). Many Bonnesen-style inequalities have been found (see [
1,
4,
12,
16,
19,
33]).
Conversely, we considered the upper bound of the isoperimetric deficit, that is,
$$ \Delta _{2} (K)=P^{2}-4\pi A\le U_{K}, $$
(1.4)
where
\(U_{K}\) is a nonnegative invariant of geometric significance, it is called the reverse Bonnesen-style inequality.
For the oval domain
K in
\(\mathbb{R}^{2}\), Bottema obtained the following reverse Bonnesen-style inequality (see [
5]):
$$ P^{2}-4\pi A\le \pi ^{2}(\rho _{M}- \rho _{m})^{2}, $$
(1.5)
where
\(\rho _{m}\) and
\(\rho _{M}\) are the minimum and maximum of the continuous curvature radius
ρ of the boundary
∂K, respectively. The equality holds if and only if
\(\rho _{m}=\rho _{M}\), that is,
K is a disc. Howard, Gao, Pan, Zhang, and others (see [
8,
17,
29]) obtained some reverse Bonnesen-style inequalities with the methods of analysis and curvature flow as follows:
$$ P^{2}-4\pi A\le c \vert \tilde{A} \vert , $$
(1.6)
where
c is a constant and
à is the area of
K̃, the domain
K̃ is bounded by the locus of the curvature centers of
∂K, where the equality sign holds if and only if
K is a disc, that is,
K̃ is a point. Some reverse Bonnesen-style inequalities for surface
\(X_{\epsilon }^{2}\) of constant curvature have been obtained in [
13,
23,
27,
28]. Zhou et al. obtained some reverse Bonnesen-style inequalities for any convex domain in [
33].
By comparing a simple closed curve and a circle, Schmidt proved the isoperimetric inequality in 1938. We were motivated by Schmidt’s works, we compared the two simple closed curves directly and obtained the symmetric mixed isoperimetric inequality (see [
14,
20,
21,
24‐
26,
30]). That is, let
\(K_{k}\) (
\(k=0,1\)) be two domains of areas
\(A_{k}\) with simple boundaries of perimeters
\(P_{k}\) in
\(\mathbb{R}^{2}\). Then
$$ P_{0}^{2}P_{1}^{2}-16\pi ^{2} A_{0}A_{1}\ge 0, $$
(1.7)
where the equality sign holds if and only if both
\(K_{0}\) and
\(K_{1}\) are discs. When one of the domains is a disc, inequality (
1.7) immediately reduces to (
1.1). That is, the symmetric mixed isoperimetric inequality (
1.7) is a generalization of the isoperimetric inequality (
1.1).
The quantity
$$ \Delta _{2}(K_{0},K_{1})=P_{0}^{2}P_{1}^{2}-16 \pi ^{2} A_{0}A_{1} $$
(1.8)
is called the symmetric mixed isoperimetric deficit of
\(K_{0}\) and
\(K_{1}\).
We were motivated by Bonnesen’s works, we considered whether there is a nonnegative invariant
\(B_{K_{0},K_{1}}\) of geometric significance such that
$$ \Delta _{2}(K_{0},K_{1})=P_{0}^{2}P_{1}^{2}-16 \pi ^{2} A_{0}A_{1}\geq B _{K_{0},K_{1}}, $$
(1.9)
where
\(B_{K_{0},K_{1}}=0\) if and only if both
\(K_{0}\) and
\(K_{1}\) are discs. An inequality of the form (
1.9) is called the Bonnesen-style symmetric mixed inequality, it is stronger than the symmetric mixed isoperimetric inequality (
1.7). Zhou, Xu, Zeng, and others (see [
14,
20,
21,
24‐
26,
30]) obtained some Bonnesen-style symmetric mixed inequalities with the known kinematic formulae of Poincaré and Blaschke.
Conversely, we considered the upper bound of the symmetric mixed isoperimetric deficit of
\(K_{0}\) and
\(K_{1}\), that is,
$$ \Delta _{2}(K_{0},K_{1})=P_{0}^{2}P_{1}^{2}-16 \pi ^{2} A_{0}A_{1}\leq U _{K_{0},K_{1}}, $$
(1.10)
where
\(U_{K_{0},K_{1}}\) is a nonnegative invariant of geometric significance, it is called the reverse Bonnesen-style symmetric mixed inequality. When one of the domains is a disc, an inequality of the form (
1.10) reduces to a reverse Bonnesen-style inequality. For any convex domain
\(K_{k}\) (
\(k=0,1\)) of areas
\(A_{k}\) and perimeters
\(P_{k}\) in
\(\mathbb{R}^{2}\), Zhou, Xu, Zeng, and others obtained the following reverse Bonnesen-style symmetric mixed inequalities (see [
21,
25,
30]):
$$\begin{aligned}& P_{0}^{2}P_{1}^{2}-16 \pi ^{2} A_{0}A_{1}\leq 4\pi ^{2}P_{0}P_{1} \bigl(R _{01}R_{1}^{2}-r_{01}r_{1}^{2} \bigr), \end{aligned}$$
(1.11)
$$\begin{aligned}& P_{0}^{2}P_{1}^{2}-16\pi ^{2} A_{0}A_{1}\le 16\pi ^{4} \bigl(R_{0} ^{2} R_{1}^{2} -r_{0}^{2} r_{1}^{2} \bigr), \end{aligned}$$
(1.12)
where
\(r_{01}=\max \{t: t(gK_{1})\subseteq K_{0}; g\in G_{2}\}\) and
\(R_{01}=\min \{t: t(gK_{1})\supseteq K_{0}; g\in G_{2}\}\) are the inradius of
\(K_{0}\) with respect to
\(K_{1}\) and the outradius of
\(K_{0}\) with respect to
\(K_{1}\), respectively.
\(G_{2}\) is a group of plane rigid motions.
\(R_{k}\) and
\(r_{k}\) are the radius of the minimum circumscribed disc and the radius of the maximum inscribed disc of
\(K_{k}\), respectively. Each equality sign holds if and only if both
\(K_{0}\) and
\(K_{1}\) are discs.
The purpose of this paper is to find some new reverse Bonnesen-style inequalities for the oval domain in
\(\mathbb{R}^{2}\), which generalize known reverse Bonnesen-style inequalities. Via the kinematic formulae of Poincaré and Blaschke, and Blaschke’s rolling theorem, we obtain a sharp reverse Bonnesen-style inequality (
3.10) in Theorem
3.2 as follows:
$$ P^{2}-4\pi A\le (2\pi \rho _{M}-P) (P-2\pi \rho _{m}), $$
which improves Bottema’s result. Furthermore, we obtain two reverse Bonnesen-style symmetric mixed inequalities (
4.10) and (
4.11) in Theorem
4.2 as follows:
$$\begin{aligned}& P_{0}^{2}P_{1}^{2}-16\pi ^{2} A_{0}A_{1}\leq 4\pi ^{2} A_{1}^{2} \bigl(\rho _{01}^{M}-\rho _{01}^{m} \bigr)^{2}, \\& P_{0}^{2}P_{1}^{2}-16\pi ^{2} A_{0}A_{1}\leq 16\pi ^{2} A_{1}^{2} \biggl(\rho _{01}^{M}- \frac{P_{0}P_{1}}{4\pi A_{1}} \biggr) \biggl(\frac{P _{0}P_{1}}{4\pi A_{1}}-\rho _{01}^{m} \biggr). \end{aligned}$$
When
\(K_{1}\) is a unit disc, (
4.10) reduces to the known reverse Bonnesen-style inequality (
1.5) of Bottema, inequality (
4.11) reduces to (
3.10).
2 Preliminaries
A set of points
K in
\(\mathbb{R}^{n}\) is convex if the line segment
\(\lambda x+(1-\lambda )y\in K\) for all
\(x, y \in K\) and
\(0 \le \lambda \le 1\). A domain is a set with nonempty interior, and an oval domain is a convex domain of boundary at least
\(C^{2}\). A convex body is a compact convex domain. The Minkowski sum of convex sets
K and
L, the scalar product of convex set
K with
\(\lambda \ge 0\) are, respectively, defined by
$$ K+L=\{x+y: x\in K, y\in L\}, $$
and
$$ \lambda K=\{\lambda x: x\in K\}. $$
A homothety of the convex set
K is of the form
\(x+\lambda K\) for
\(x\in \mathbb{R}^{n}\),
\(\lambda > 0\).
For the proof of the main theorem, we cite Blaschke’s rolling theorem in
\(\mathbb{R}^{2}\) from [
3,
7,
13,
25].
By Lemma
2.1, we obtain the following corollary.
3 Reverse Bonnesen-style inequalities
Let
K be an oval domain of area
A and perimeter
P in
\(\mathbb{R}^{2}\). Let
\(\rho (\partial K)\) be the curvature radius of boundary
∂K and
\(\rho _{m}=\min \{\rho (\partial K)\}\),
\(\rho _{M}=\max \{\rho (\partial K)\}\). Let
dg denote the kinematic density of the group
\(G_{2}\) of plane rigid motions, and
\(B_{t}\) be a circle of radius
t in
\(\mathbb{R}^{2}\). Let
\(n\{\partial K\cap \partial (gB_{t})\}\) denote the number of points of intersection
\(\partial K\cap \partial (gB_{t})\) and
\(\chi \{K\cap gB_{t}\}\) be the Euler–Poincaré characteristics of the intersection
\(K\cap gB_{t}\). Then we have the following kinematic formula of Poincaré (see [
18]):
$$ \int _{\{g\in G_{2}: \partial K\cap \partial (gB_{t})\ne \emptyset \}} n\bigl\{ \partial K\cap \partial (gB_{t}) \bigr\} \,dg=8\pi Pt $$
(3.1)
and the kinematic formula of Blaschke
$$ \int _{\{g\in G_{2} : K\cap gB_{t}\ne \emptyset \}} \chi \{K\cap gB _{t}\}\,dg=2\pi ^{2}t^{2}+2\pi Pt+2\pi A. $$
(3.2)
If
μ denotes a set of all positions of
\(B_{t}\) in which either
\(gB_{t}\subset K\) or
\(gB_{t}\supset K\), then the kinematic formula of Blaschke (
3.2) can be rewritten as
$$ \int _{\mu }dg=2\pi ^{2}t^{2}+2\pi Pt+2\pi A- \int _{\{g\in G_{2} : \partial K\cap \partial (gB_{t})\neq \emptyset \}} \chi \{K\cap gB_{t}\}\,dg. $$
(3.3)
Since
K is an oval domain in
\(\mathbb{R}^{2}\), then
$$ \int _{\{g\in G_{2} : \partial K\cap \partial (gB_{t})\neq \emptyset \}} \chi \{K\cap gB_{t}\}\,dg= \int _{\{g\in G_{2} : \partial K\cap \partial (gB_{t})\neq \emptyset \}}\,dg. $$
(3.4)
When
\(t\in (0,\rho _{m}]\) or
\(t\in [\rho _{M}, +\infty )\), by Corollary
2.1, we have
\(n\{\partial K\cap \partial (gB_{t})\}=2\) or
\(gB_{t}\) is tangent to
∂K. When
\(gB_{t}\) is tangent to
∂K, we have
$$ \int _{\{g\in G_{2}: \partial K\cap \partial (gB_{t})\ne \emptyset \}} n\bigl\{ \partial K\cap \partial (gB_{t}) \bigr\} \,dg=0, $$
(3.5)
therefore,
$$ \int _{\{g\in G_{2}: \partial K\cap \partial (gB_{t})\ne \emptyset \}} n\bigl\{ \partial K\cap \partial (gB_{t}) \bigr\} \,dg= \int _{\{g\in G_{2}: \partial K\cap \partial (gB_{t})\ne \emptyset \}} 2dg. $$
(3.6)
By (
3.4) and (
3.6), we have
$$ \int _{\{g\in G_{2} : \partial K\cap \partial (gB_{t})\neq \emptyset \}} \chi \{K\cap gB_{t}\}\,dg= \frac{1}{2} \int _{\{g\in G_{2}: \partial K\cap \partial (gB_{t})\ne \emptyset \}} n\bigl\{ \partial K\cap \partial (gB_{t}) \bigr\} \,dg. $$
(3.7)
Therefore, when
\(t\in (0,\rho _{m}]\) or
\(t\in [\rho _{M}, +\infty )\), by (
3.3), (
3.7), and (
3.1), we obtain
$$\begin{aligned} \int _{\mu }dg =& 2\pi ^{2}t^{2}+2\pi Pt+2 \pi A- \int _{\{g\in G_{2} : \partial K\cap \partial (gB_{t})\neq \emptyset \}} \chi \{K\cap gB_{t}\}\,dg \\ =& 2\pi ^{2}t^{2}+2\pi Pt+2\pi A-\frac{1}{2} \int _{\{g\in G_{2}: \partial K\cap \partial (gB_{t})\ne \emptyset \}} n\bigl\{ \partial K\cap \partial (gB_{t}) \bigr\} \,dg \\ =& 2\pi ^{2}t^{2}+2\pi Pt+2\pi A-4\pi Pt \\ =& 2\pi ^{2}t^{2}-2\pi Pt+2\pi A \\ \ge & 0. \end{aligned}$$
(3.8)
For all
\(a\geq 0\),
\(b\geq 0\), we have
\(4ab\leq (a + b)^{2}\), that is,
$$ (2\pi \rho _{M}-P) (P-2\pi \rho _{m})\leq \pi ^{2}(\rho _{M}-\rho _{m})^{2}. $$
Therefore, the upper bound of the isoperimetric deficit in inequality (
3.10) is better than the upper bound in inequality (
1.5), that is, the reverse Bonnesen-style inequality (
3.10) strengthens Bottema’s result.
4 Reverse Bonnesen-style symmetric mixed inequalities
Let
\(K_{k}\) (
\(k = 0, 1\)) be two oval domains in
\(\mathbb{R}^{2}\). Let
\(\rho (\partial K_{k})\) be the curvature radii of boundaries
\(\partial K_{k}\), and let
\(\rho _{m}(\partial K_{k})=\min \{\rho ( \partial K_{k})\}\),
\(\rho _{M}(\partial K_{k})=\max \{\rho (\partial K _{k})\}\). Let
$$ \rho _{m}^{g}(K_{0}, K_{1})=\max \bigl\{ t: \rho _{M}\bigl(\partial \bigl(t(gK_{1})\bigr)\bigr) \leq \rho _{m}(\partial K_{0}); g\in G_{2} \bigr\} $$
and
$$ \rho _{M}^{g}(K_{0}, K_{1})=\min \bigl\{ t: \rho _{m}\bigl(\partial \bigl(t(gK_{1})\bigr)\bigr) \geq \rho _{M}(\partial K_{0}); g\in G_{2} \bigr\} $$
be the inradius and the outradius of curvature,
\(K_{0}\) with respect to
\(K_{1}\), where
\(G_{2}\) is a group of plane rigid motions. It is obvious that
\(\rho _{m}^{g}(K_{0}, K_{1})\leq \rho _{M}^{g}(K_{0}, K_{1})\). Since both
\(\rho _{m}^{g}(K_{0}, K_{1})\) and
\(\rho _{M}^{g}(K_{0}, K_{1})\) are rigid invariant, we simply denote them by
\(\rho _{01}^{m}\) and
\(\rho _{01}^{M}\), respectively. Note that, if
\(K_{1}\) is a unit disc, then
\(\rho _{01}^{m}\) and
\(\rho _{01}^{M}\) are the minimum
\(\rho _{m}( \partial K_{0})\) and the maximum
\(\rho _{M}(\partial K_{0})\) of the continuous curvature radius of the boundary
\(\partial K_{0}\), respectively.
Let
\(K_{k}\) (
\(k=0,1\)) be two oval domains of areas
\(A_{k}\) and perimeters
\(P_{k}\) in
\(\mathbb{R}^{2}\). Let
dg denote the kinematic density of the group
\(G_{2}\) of plane rigid motions. Let
\(n\{\partial K_{0} \cap \partial (t(gK_{1}))\}\) denote the number of points of intersection
\(\partial K_{0}\cap \partial (t(gK_{1}))\), and let
\(\chi \{K_{0} \cap t(gK_{1})\}\) be the Euler–Poincaré characteristics of the intersection
\(K_{0}\cap t(gK_{1})\). Then we have the following kinematic formula of Poincaré (see [
14,
20,
21,
24‐
26,
30]):
$$ \int _{\{g\in G_{2}: \partial K_{0}\cap \partial (t(gK_{1}))\ne \emptyset \}} n\bigl\{ \partial K_{0}\cap \partial \bigl(t(gK_{1})\bigr)\bigr\} \,dg=4tP_{0}P_{1} $$
(4.1)
and the kinematic formula of Blaschke
$$ \int _{\{g\in G_{2} : K_{0}\cap t(gK_{1})\ne \emptyset \}} \chi \bigl\{ K _{0}\cap t(gK_{1})\bigr\} \,dg=2\pi \bigl(t^{2}A_{1}+A_{0} \bigr)+tP_{0}P_{1}. $$
(4.2)
Let
μ denote a set of all positions of
\(K_{1}\) in which either
\(t(gK_{1})\subset K_{0}\) or
\(t(gK_{1})\supset K_{0}\), then (
4.2) can be rewritten as
$$ \int _{\mu }dg=2\pi \bigl(t^{2}A_{1}+A_{0} \bigr)+tP_{0}P_{1}- \int _{\{g\in G_{2} : \partial K_{0}\cap \partial (t(gK_{1}))\neq \emptyset \}} \chi \bigl\{ K_{0}\cap t(gK_{1})\bigr\} \,dg. $$
(4.3)
Since
\(K_{k}\) (
\(k=0,1\)) are two oval domains in
\(\mathbb{R}^{2}\), then
$$ \int _{\{g\in G_{2} : \partial K_{0}\cap \partial (t(gK_{1}))\neq \emptyset \}} \chi \bigl\{ K_{0}\cap t(gK_{1})\bigr\} \,dg= \int _{\{g\in G_{2} : \partial K_{0}\cap \partial (t(gK_{1}))\neq \emptyset \}}\,dg. $$
(4.4)
When
\(t\in (0, \rho _{01}^{m}]\) or
\(t\in [\rho _{01}^{M}, +\infty )\), we can obtain
\(\rho _{M}(\partial (t(gK_{1})))\leq \rho _{m}(\partial K _{0})\) or
\(\rho _{m}(\partial (t(gK_{1})))\geq \rho _{M}(\partial K_{0})\). By Corollary
2.2, we have
\(n\{\partial K_{0}\cap \partial (t(gK_{1}))\}= 2\) or
\(\partial (t(gK_{1}))\) is tangent to
\(\partial K _{0}\). When
\(\partial (t(gK_{1}))\) is tangent to
\(\partial K_{0}\), we have
$$ \int _{\{g\in G_{2}: \partial K_{0}\cap \partial (t(gK_{1}))\ne \emptyset \}} n\bigl\{ \partial K_{0}\cap \partial \bigl(t(gK_{1})\bigr)\bigr\} \,dg=0, $$
(4.5)
therefore,
$$ \int _{\{g\in G_{2}: \partial K_{0}\cap \partial (t(gK_{1}))\ne \emptyset \}} n\bigl\{ \partial K_{0}\cap \partial \bigl(t(gK_{1})\bigr)\bigr\} \,dg= \int _{\{g\in G_{2}: \partial K_{0}\cap \partial (t(gK_{1}))\ne \emptyset \}} 2 dg. $$
(4.6)
By (
4.4) and (
4.6), we have
$$\begin{aligned}& \int _{\{g\in G_{2} : \partial K_{0}\cap \partial (t(gK_{1}))\neq \emptyset \}} \chi \bigl\{ K_{0}\cap t(gK_{1})\bigr\} \,dg \\& \quad =\frac{1}{2} \int _{\{g\in G_{2} : \partial K_{0}\cap \partial (t(gK_{1}))\neq \emptyset \}} n\bigl\{ \partial K_{0}\cap \partial \bigl(t(gK_{1})\bigr)\bigr\} \,dg. \end{aligned}$$
(4.7)
Therefore, when
\(t\in (0, \rho _{01}^{m}]\) or
\(t\in [\rho _{01}^{M}, + \infty )\), by (
4.3), (
4.7), and (
4.1), we obtain
$$\begin{aligned} \int _{\mu }dg =& 2\pi \bigl(t^{2}A_{1}+A_{0} \bigr)+tP_{0}P_{1}- \int _{\{g\in G_{2} : \partial K_{0}\cap \partial (t(gK_{1}))\neq \emptyset \}} \chi \bigl\{ K_{0}\cap t(gK_{1})\bigr\} \,dg \\ =& 2\pi \bigl(t^{2}A_{1}+A_{0} \bigr)+tP_{0}P_{1}-\frac{1}{2} \int _{\{g\in G_{2} : \partial K_{0}\cap \partial (t(gK_{1}))\neq \emptyset \}} n\bigl\{ \partial K_{0}\cap \partial \bigl(t(gK_{1})\bigr)\bigr\} \,dg \\ =& 2\pi A_{1}t^{2}-P_{0}P_{1}t+2\pi A_{0} \\ \ge & 0. \end{aligned}$$
(4.8)
When
\(K_{1}\) is a unit disc, inequality (
4.9) immediately reduces to inequality (
3.9).
We now obtain the following reverse Bonnesen-style symmetric mixed inequalities.
When
\(K_{1}\) is a unit disc, the reverse Bonnesen-style symmetric mixed inequality (
4.10) immediately reduces to the known reverse Bonnesen-style inequality (
1.5) of Bottema, inequality (
4.11) reduces to inequality (
3.10). For all
\(a\geq 0\),
\(b\geq 0\), we have
\(4ab\leq (a + b)^{2}\), that is,
$$ 16\pi ^{2} A_{1}^{2} \biggl(\rho _{01}^{M}- \frac{P_{0}P_{1}}{4\pi A_{1}} \biggr) \biggl( \frac{P_{0}P_{1}}{4\pi A _{1}}-\rho _{01}^{m} \biggr)\leq 4\pi ^{2} A_{1}^{2} \bigl(\rho _{01} ^{M}-\rho _{01}^{m} \bigr)^{2}. $$
Therefore, the upper bound of the symmetric mixed isoperimetric deficit in inequality (
4.11) is better than the upper bound in inequality (
4.10), that is, the reverse Bonnesen-style symmetric mixed inequality (
4.11) is stronger than inequality (
4.10).
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